SOME ISSUES IN THE THEORY OF SUPERNILPOTENT RADICALS S. Tumurbat University of Mongolia, P.O. Box 75 Ulaan Baatar 20, Mongolia C

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1 SOME ISSUES IN THE THEORY OF SUPERNILPOTENT RADICALS S. Tumurbat University of Mongolia, P.O. Box 75 Ulaan Baatar 20, Mongolia Currently at A. Rényi Institute of Mathematics, Hungarian Academy of Sciences P.O. Box 127, H 1364 Budapest, Hungary tumurbat@renyi.hu Dedicated to the memory of ProfessorV.A.Andrunakievich Contents 1. Kurosh Amitsur radicals and definitions 2. Weakly special classes and supernilpotent radicals 3. Special classes and special radicals 4. Coincidence of special radicals 5. On subhereditary radicals 6. Köthe's problem and its approximation by radicals 7. Morita context, normal radicals, N-radicals 8. On polynomial and multiplicative radicals References 1. Kurosh Amitsur radicals and definitions In this paper rings are associative, not necessarily with a unity. As usual, I/A, L/ l A and R/ r A will denote that I is an ideal, L is a left ideal and R is a right ideal in a ring A, respectively. Let us recall that a(kurosh Amitsur) radical fl is a class of rings which is closed under homomorphisms, extension (I and A=I in fl imply A 2 fl), and has the inductive property (ifi 1 I ::: is a chain of ideals of the ring A = [I and each I is in fl, thena is in fl). Every ring A contains a unique largest fl-ideal (i.e. an ideal which is in fl) denoted by fl(a), which is called the fl-radical of A. If fl is a radical, the class Sfl = fa j fl(a) =0g 2000 Mathematics Subject Classification: 16N80. The author gratefully acknowledges the support of the Hungarian OTKA Grant # T Typeset by AMS-TEX 1

2 2 S. Tumurbat is called the semisimple class of fl. A class M of rings is a hereditary class meaning that I/A2Mimplies I 2M. A class M of rings is said to be regular if every non-zero ideal of a ring in M has a non-zero homomorphic image in M. Starting from a regular (in particular, hereditary) class M of ring the upper radical operator U yields a radical class: UM = fa j A has no non-zero homomorphic image in Mg: The fundamental properties of radicals can be found in [5; 19; 75; 88]. A radical fl containing all nilpotent rings is called a hypernilpotent radical. A radical fl is said to be supernilpotent if it is hereditary and hypernilpotent. We list some important examples of supernilpotent radicals. ffl The Baer radical fi. This is the upper radical determined by the class of all semiprime rings. ffl The locally nilpotent radical L. This is the radical class of all locally nilpotent rings. ffl The Jacobson radical J. This is the upper radical determined by the class of all primitive rings. ffl The Behrens radical B. This is the upper radical determined by the class of all subdirect irreducible rings possessing a non-zero idempotent in the heart. ffl The Brown McCoy radical G. This is the upper radical determined by the class of all simple rings with unity. It is well known that fi;l; J ; B and G are supernilpotent radicals. ffl The Thierrin radical. This is the upper radical determined by the class of all division rings. ffl The antisimple radical. This is the upper radical determined by the class of all subdirectly irreducible rings with idempotent hearts. The ring A with unity adjoined will be denoted by A 1. A (left) ideal L of a ring A is said to be essential if I L 6= 0, for every non-zero ideal I of A. Denote by (L/ l A) L/ A, respectively. A right essential ideal is defined by the dual condition. A subring S of A is said to be accessible if S = I 1 /I 2 / /I n = A for some natural number n. A class M of rings is said to be left (right) hereditary if L/ l A 2M(R/ r A 2 M), then L 2M(R 2M), respectively. We denote by N the class of all nil rings. The upper radical N s is determined by the class of rings which contain no nonzero nil left ideals. A radical fl is called principally left hereditary if A 2 fl implies Aa 2 fl for every element a 2 A. Principally right hereditary radicals may be defined similarly. A radical fl is called left strong, if L/ l A and L 2 fl implies L fl(a). Right strong radicals may be defined similarly. The compiled material focuses to results on which also the author has worked in the recent years. In the presentation we followed the terminology and notation of the book [31], which we consider as a sourse of further reference.

3 Some issues in the theory of supernilpotent radicals 3 2. Weakly special classes and supernilpotent radicals If M is a regular class of semiprime rings, then its upper radical UM will contain all zero-rings. This upper radical need not be hereditary even, if M is hereditary. For example, M = fall idempotent simple ringsg: Clearly M is a hereditary class, but UM is not hereditary. For any class % of rings, we define the essential cover operator E by E% = fa j there is a ring I 2 % such thati/ Ag; that is, E% is the class of all essential extensions of rings from %. This class E% is called the essential cover of %. This essential extension plays an important role in guaranteeing that an upper radical is hereditary. Proposition 2.1. (Anderson and Wiegandt [1]). If % is a regular class of semiprime rings, then so is its essential cover E%. Proposition 2.2. (le Roux, Heyman and Jenkins [64]). If % is a regular class of semiprime rings and E% its essential cover, then the upper radical UE% is hereditary and therefore supernilpotent. We shall say that the radical class fl has the intersection property relative to the class %, if fl(a) =(A) % for every ring A: Recall that (A) % = (I /Aj A=I 2 %): Theorem 2.3. (Rjabukhin [60] and le Roux, Heymann and Jenkins [64]). For a regular class % of semiprime rings the following are equivalent: (i) U% is hereditary, (ii) U% has the intersection property relative to the essential cover E% of %. Corollary 2.4. Let % be a regular class of semiprime rings. If fl = U% is a supernilpotent radical, then every fl-semisimple ring A 2 Sfl is a subdirect sum A = X subdirect (A ff j A ff 2E%): Example 2.5. The Brown McCoy radical G, the Jacobson radical J, the Behrens radical B, the locally nilpotent radical L, and the Baer radical fi are hereditary. Hence all simple rings with unity, all primitive rings, all subdirect reducible ring with idempotents in the heart, all semiprime rings have no locally nilpotent ideals and all prime rings satisfy the conditions of Theorem 2.3 In general, the essential cover E% of a class of rings, is not hereditary and also not closed under essential extensions.

4 4 S. Tumurbat Proposition 2.6 (Heyman and Roos [33]). If % is a hereditary class of semiprime rings, then its essential cover E% is also a hereditary class of semiprime rings. If % is a hereditary class of rings, then E% is a closed essential extension. We define the hereditary closure operator H by H % = fa j A is an accessible subring of a ring B 2 %g: Theorem 2.7 (Gardner and Wiegandt [31]).. Let % be any class of semiprime rings, and let ff be the class of all subdirect sums of rings in EH %. Then ff is the smallest semisimple class containing % such that Uff = UEH % is supernilpotent. A hereditary class % of semiprime rings is said to be weakly special if I/A, I 2 % and Ann(I) =fa 2 A j Ia = ai =0g =0; then A 2 %. Theorem 2.8 (Rjabukhin [60]). If % is a weakly special class, then the upper radical fl = U% is supernilpotent and has the intersection property relative to %. Every supernilpotent radical is the upper radical of a weakly special class. The class of all semiprime rings is a weakly special class. In fact, the largest weakly special class. 3. Special classes and special radicals We begin with the following results on prime rings. Theorem 3.1. A ring A is semiprime if and only if the intersection of all the prime ideals of A is zero. Lemma 3.2. (i) If I/Aand A is a prime ring, then so is I. (ii) If I/ A and I isaprimering,thensoisa. (iii) The class P of all prime rings is a weakly special class. Andrunakievich [3; 4] called a class % of prime rings a special class if % is hereditary and satisfies the following: if I/A; I 2 % and Ann(I) =0; then A 2 %: Clearly, every special class is a weakly special class. A radical fl will be called a special radical, iffl is the upper radical of a special class. Examples. (i) P is a special class (the largest special class). (ii) The class of all primitive rings is a special class. (iii) The class of subdirectly irreducible rings with idempotent heart is a special class. (iv) The class of all simple rings with unity is a special class. Thus the Baer radical fi, the Jacobson radical J, the Behrens radical B and the Brown McCoy radical G are special radicals. It is clear that every special radical is supernilpotent and hence hereditary.

5 Some issues in the theory of supernilpotent radicals 5 Corollary 3.3. Let % be aspecial class of rings and fl = U%. If A is a fl-semisimple ring, then A is a subdirect sum of rings in %. Theorem 3.4 (Gardner and Wiegandt [31]. Let fl be a hereditary radical. Then the class Sfl P of prime fl-semisimple rings is a special class and it is a largest special class contained insfl. The class fl isaspecial radical if and only if fl = U(Sfl P). Corollary 3.5. If % is a hereditary class of prime rings, then E% is the smallest special class containing %. In particular, if % is a class of simple prime rings, then the smallest special class containing % is the class of all subdirectly irreducible rings each having its heart in %. The following theorem gives a characterization of a class of rings which is a semisimple class of a special radical. Theorem 3.6 (Rjabukhin and Wiegandt [62]). A class % of rings is the semisimple class of a special radical if and only if (i) % is regular, (ii) % is closed under subdirect sums, (iii) % is closed under essential extensions, (iv) if A 2 %, then A is a subdirect sum of ring A ff,each in %. A = X subdirect (A ff j A ff 2 % P ) The following theorem gives a characterization of special radical classes. AringB is said to be a prime image ofaringa, ifb is a homomorphic image of A and B is a prime ring. Theorem 3.7 (Gardner and Wiegandt [30]). A class fl of rings is a special radical if and only if (i) fl is homomorphically closed, (ii) fl is hereditary, (iii) if every non-zero prime image of a ring A has a non-zero ideal in fl, then A is in fl. Let R be a commutative ring with unity, W be the subvariety of the variety of all R-algebras, fl be a radical in W and P be the abstract subclass of all prime rings in W. We denote by Sd(P) the subclass of all subdirect products of algebras of P in W, K P is the class of all algebras A 2 W, having ideals C/B/A, such that 0 6= B=C 2Pand I B 6 C, for any non-zero ideals I of A, and Var(P) isthe subvariety of the variety W, generated by class P. Theorem 3.8 (Beidar [7]). Let fl be a radical with the intersection property relative to the class P. Then K P Var(P) Sd(P).

6 6 S. Tumurbat Theorem 3.9 (Beidar [7]). Let fl be aradical in W, andp be anabstract class of prime rings in W. Suppose that K P Var(P). Then the following conditions are equivalent (i) fl has the intersection property relative to the class P. (ii) The class P satisfies the following conditions: a) P is a regular class; b) K P Sd(P); c) P Sfl; d) there exists an ideal B of A such that 0 6= A=B 2P for any 0 6= A 2 Sfl. Corollary Let W be the variety of all associative algebras. Then the following are equivalent: (i) fl is a special radical, (ii) fl has the intersection property relative to a class of prime algebras. The smallest supernilpotent radical containing a given hereditary and homomorphically closed class ffi is given simple as the lower radical L(ffi [ Z), where Z is the class of all zero-rings. If we again start with a given class ffi, it is not quite so easy to construct the smallest special radical containing ffi. The solution is based on the characterization given in Theorem 3.7. Theorem 3.11 (Gardner and Wiegandt [30]). Let ffi be a hereditary and homomorphically closed classofrings. Then the smallest special radical containing ffi is the class ρ fi ff L sp fi if every non-zero prime image of ffi = Afi : a ring A has a non-zero ideal in ffi We consider the class ff of prime rings such thatevery non-zero prime image has a non-zero zero divisors. The class % of all domains (that is, of all rings without zero divisors) is a special class. ff is hereditary and homomorphically closed. By the above theorem L sp ff is a special radical (smallest) containing ff. Hence the generalized nil radical (that is, N g = U%) is generated by ff, as a special radical. We know that many radicals are special radical. However, there are supernilpotent radicals which are not special. Example 3.12 (Rjabukhin [60]). Let us consider the the class K = fall Boolean rings A j every non-zero ideal of A is infiniteg: Then the upper radical UK is not a special and supernilpotent. More examples of non-special supernilpotent radicals were given by van Leeuwen and Jenkins [87], Rjabukhin [61], Beidar and Salavová [16], Gardner and Stewart [29], France Jackson [26] and Beidar and Wiegandt [18]. The characteristic difference which makes a supernilpotent radical to a special one is given in the following. Theorem 3.13 (Beidar [7]). Let fl be a radical. fl is a special radical if and only if fl has the intersection property relative to the class % = Sfl P. If fl is any supernilpotent and non-special radical, then there exists a smallest special radical fl 0 containing fl. This special radical fl 0 can be obtained as fl 0 = U(Sfl P )orfl 0 = L sp fl.

7 Some issues in the theory of supernilpotent radicals 7 4. Coincidence of special radicals It is well known that for any partition ( ; ) of simple rings the lower radical L is properly contained in the upper radical U (see for instance Divinsky [19] and Szasz [75]). Miguel Ferrero posed the question as whether the lower special radical of (which is much bigger than L ) differs from the dual special radical determined by (which may be considerably smaller than U ) even if we impose further restrictions. Tumurbat and Wiegandt [81] gave an affirmative answer to this question. It can be achieved that there is a polynomial ring which distinguishes these two radical classes. If fi is a special class of subdirectly irreducible rings, the upper radical Ufi is said to be a dual special radical. Let C denote the class of all simple prime rings. Proposition 4.1 (Tumurbat and Wiegandt [81]). The class % = P SL(C [ fi) coincides with the class of all prime rings having no minimal ideals. Proposition 4.2 (Tumurbat and Wiegandt [81]). If fl and ffi are special radicals such that (i) fl C = ffi C, (ii) fl % = ffi %, then fl = ffi. We consider the class s( ) =fa 2 P j A is subdirectly irreducible with heart H(A) 2 g: Proposition 4.3 (Tumurbat and Wiegandt [81]). P SL sp = % [ s( ). The following theorem shows that the lower special radical of the class is strictly contained in the dual special radical determined by the special class s( ). Crucial is the existence of prime rings which have no non-zero minimal ideals and no proper prime images, or equivalently, which are not subdirectly irreducible and have no proper prime images. Such rings are referred to as Λ-rings (cf. [25]). We recall that a prime ring A is said to be a Λ-ring if any proper prime image A 0 of A is in fi. We consider the following examples, which is needed in the proof of the following Theorems 4.4; 4.8 and Examples. (i) The non-nil Jacobson radical ring n 2x J = 2y +1 fi fi fi x; y 2 Z; and (2x; 2y +1)=1 is an example of such a ring, as observed in [25] (cf. also [19, Example 10 on p. 103]. (ii) Let M(A) denote the ring of all infinite matrices over A having only finitely many non-zero elements. If A is a Λ-ring, then M(A) is a Λ-ring with trivial center and if A has no non-zero minimal ideals, then M(A) has no non-zero minimal ideals. Now, we shall give a proof of (ii). First, we showthattoany non-zero ideal K of M(A), there exists a non-zero ideal I of A such that M(I) K. Consider a o

8 8 S. Tumurbat non-zero matrix B =(b ij ) 2 K. We may assume that b ij 6= 0 for some fixed indices i and j. Let (A) uν denote the subset of M(A) having non-zero elements only at the (u; ν) entry. For arbitrary indices k and l we have (A) ki B(A) jl = 0 0 ::: 0 ::: 0 1 C.. 0 ::: Ab ij A ::: 0 C.. A (A) kl K: 0 ::: 0 Since Λ-rings are prime rings, I = Ab ij A 6= 0. Obviously I/Asuch that M(I) = X k;l (I) kl K and M(I) /M(A). Next, we prove thatm(a)=k is not a prime ring whenever M(A) 6= K 6= 0. Now, take anon-zero ideal I of A such thatm(i) K. Since A is a Λ-ring, we have A=I 2 fi. Hence also M(A)=M(I) ο = M(A=I) 2 fi and therefore, taking into account that M(I) K, alsom(a)=k 2 fi follows. Since A is a prime ring, so is M(A). Since M(A) consists of matrices having only finitely many non-zero elements, M(A) does not contain non-zero diagonal elements, whence the center of M(A) is 0. Suppose that K is a non-zero minimal ideal of M(A). We maychoose a non-zero ideal I of A such that M(I) K. Since A has no non-zero minimal ideals, there exists a non-zero ideal H of A such that 0 6= H ρ I. Hence 0 6= M(H) ρ M(I) K and M(H) /M(A), a contradiction. Thus M(A) has no non-zero minimal ideals. We setff = % [ s( ). Theorem 4.4 (Tumurbat and Wiegandt [81]). L sp = Uff $ Us( ): Let us consider the partition ( ; ) of simple prime rings where = C J. Further, set J ffi = Us( ) for the upper radical of all subdirectly irreducible rings with Jacobson semisimple heart. Proposition 4.5 (Tumurbat and Wiegandt [81]). L sp $ J $ J ffi:

9 Some issues in the theory of supernilpotent radicals 9 Corollary 4.6. Let fl be a special radical and = fl C. Then fl = L sp if and only if % Sfl. In Theorem 4.4 we have seen that for the largest special class ff determining L sp,wehave Uff ρus( ). The next theorem gives a criterion on a special class fi containing s( ) such that Ufi = Us( ). Theorem 4.7 (Tumurbat and Wiegandt [81]). Let fi be a special class such that s( ) fi ρ ff holds. Ufi = Us( ) if and only if for every hereditary subclass fi 0 6= f0g of fi the class Lfi 0 s( ) contains a non-zero ring. Theorem 4.8 (Tumurbat and Wiegandt [81]). If ( ; ) is a partition of simple rings such that the class does not contain countable simple rings without unity element, then there exists a ring B such that B[x] 2Us( ) nl sp. Corollary 4.9. If the class consists of simple rings A with unity element or jaj 0, then there exists a ring B such that B[x] 2Us( ) nl sp. Theorem 4.10 (Tumurbat and Wiegandt [81]). Let ( ; ) be a partition of simple rings. If the class does not contain countable fields of characteristic 0, then there exists a ring A such that A[x] 2Us( ) nl sp. Corollary Let ( ; ) be a partition of simple prime rings such that the class either does not contain countable simple rings without unity element, or does not contain countable fields of characteristic 0. Further, consider any special radical fl. (i) If fl(a[x]) = L sp (A[x]) for all rings A, thenfl 6= Us( ). (ii) If fl(a[x]) = Us( )(A[x]) for all rings A, thenfl 6= L sp. Miguel Ferrero asked also the following question (cf. [81]): Let fl and ffi be special radicals such that (i) fl(a) =ffi(a) for all simple rings A; (ii) fl(a[x]) = ffi(a[x]) for all rings A. Does then fl coincide with ffi? In [80], a negative answer was given to this question and also a criterion for the coincidence. We say that a radical fl has the Amitsur property if fl(a[x]) = (A fl(a[x])[x] for every ring A: Let fl be a special radical. Let fl 0 denote the lower hereditary radical generated by the class of all polynomial rings which are in fl. Clearly fl 0 fl. A prime ring A 2 fl is said to be O fl -ring for a special radical fl if (A): A has no minimal ideals and which arefl 0 -semisimple, (B): A has no non-zero accessible subring B such that B ο = In / /I 1 = C[x]; for a ring C: We set Clearly, O fi = O L = ;. O fl = fa 2 fl j A is a O fl -ringg:

10 10 S. Tumurbat Theorem 4.12 [80]. Let fl be a special radical which has the Amitsur property. If there exists a non-zero prime B with condition (A), then there exists a special radical ffi such that (i) ffi ρ fl; (ii) fl C = ffi C; (iii) fl(a[x]) = ffi(a[x]), for any ring A; (iv) O fl 6= ;. Theorem 4.13 [80]. There exists a special radical ffi such that (i) ffi ρj, (ii) J C = ffi C; (iii) J (A[x]) = ffi(a[x]), for any ring A. Corollary The non-nil Jacobson radical ring J is in O J. The next theorem gives a criterion for the coincidence of two special radicals. Theorem 4.15 [80]. Let fl and ffi be special radicals such that (i) fl C = ffi C; (ii) fl(a[x]) = ffi(a[x]) for all rings A. Then fl = ffi if and only if O fl = O ffi. Corollary Let ffi fl be special radicals such that (i) fl C = ffi C; (ii) fl(a[x]) = ffi(a[x]), for all rings A. Then fl = ffi if and only if O fl ffi. Let us consider the class of radical classes O = ffl jo fl = ; and fl is a special radicalg: Corollary Let fl and ffi be special radicals in O such that (i) fl C = ffi C; (ii) fl(a[x]) = ffi(a[x]) for all rings A. Then ffi = fl. Gardner posed the following question (cf. [26]): Do there exist disjoint special classes M 1 and M 2 of prime rings with UM 1 = UM 2? Beidar gave an affirmative answer to this question. We denote by Sp(A), the special class generated by the ring A. Lemma 4.18 (Beidar [8]). If % and ff are classes of prime rings and % Sd(ff), ff Sd(%), then USp(%) =USp(ff). This Lemma is used in the proof of the following theorem. Theorem 4.19 (Beidar [8]). Let C be an algebraic closure of the field Q of rational numbers, A = C [x 1 ;x 2 ;:::] be the C -algebra of polynomials over C in commuting indeterminates x 1 ;x 2 ;:::, I be thec-ideal of A generated by the polynomial x x 2 2 1, andb = A=I. Then (i) USp(A) =USp(B); (ii) Sp(A) Sp(B) =f0g.

11 Some issues in the theory of supernilpotent radicals On subhereditary radicals Puczyψlowski and Zand [58] called a radical fl left subhereditary if 0 6= L/ l A 2 fl implies fl(l) 6= 0: A right subhereditary radical is defined by the dual condition. Proposition 5.1 (Puczyψlowski and Zand [58]). Given a radical fl the following are equivalent. (i) fl is left subhereditary; (ii) non-zero fl-radical rings contain no essential fl-semisimple left ideals; (iii) L/ l A and L 2 Sfl imply A 2 Sfl. Clearly, a left hereditary radical is a left subhereditary radical. Proposition 5.2 (Puczyψlowski and Zand [58]). The generalized nil radical N g is not left subhereditary. Proposition 5.3 (Puczyψlowski and Zand [58]). The upper radical UC r determined by the class C r of commutative reduced rings is left subhereditary but is not left hereditary. This proposition shows that left subhereditary radicals are a proper generalization of left hereditary radicals. Puczyψlowski and Zand [58] also investigated subhereditariness of lower radicals and the lattice of subhereditary radicals. Proposition 5.4 (Puczyψlowski and Zand [58]). Given a homomophically closed class M of rings, the lower radical LM is a left subhereditary radical if and only if for every 0 6= L/ l A with A 2M, we have LM(L) 6= 0. In particular we get Corollary 5.5. If M is a homomorphically closed class of rings such that every non-zero left ideal of each ring in M contains a non-zero ideal in M, then the radical LM is left subhereditary. Applying Corollary 5.5 it is easy to see that if fl ff are left subhereditary radicals, then so fl = L([fl ff ). Hence the class of left subhereditary radicals is a join subsemilattice of the lattice of all radicals. The following proposition shows that actually it is a sublattice of that lattice. Proposition 5.6 (Puczyψlowski and Zand [58]). If fl and ffi are left subhereditary radicals, then so is fl ffi. Now we shall obtain some results on the structure of the lattice of left subhereditary radicals. A radical consisting of idempotent rings is called hypoidempotent. Proposition 5.7 (Puczyψlowski and Zand [58]). Every left subhereditary and hypoidempotent radical fl consists of reduced rings. Given a prime p, let Z 0 p denote the ring with zero multiplication on an additive group of order p. Given a simple ring A, lete(a) =fl/ l A j L 2 = Lg.

12 12 S. Tumurbat Proposition 5.8 (Puczyψlowski and Zand [58]). If fl is a left subhereditary radical, then L(Z 0 p) fl, for a prime p or L(E(A)), for a simple domain A. Proposition 5.9 (Puczyψlowski and Zand [58]). Let A be a simple idempotent ring. Then (i) the radical L(E(A)) is left subhereditary if and only if A is a domain; (ii) the radical L(E(A)) is left hereditary if and only if A is a division ring; (iii) the radical L(fi [ E(A)) is left hereditary. From Propositions 5.8 and 5.9 it follows that if fl is an atom in the lattice of left subhereditary radicals, then fl = L(Z 0 p)foraprimep or fl = L(E(A)) for a simple domain A. It is easy to see that all radicals L(Z 0 p) are left subhereditary. Since L(Z 0 p) is an atom in the lattice of hereditary radicals, all L(Z 0 p) are atoms in the lattice of left subhereditary radicals. Proposition 5.10 (Puczyψlowski and Zand [58]). For a simple domain A the following are equivalent: (i) LE(A) is an atom in the lattice of left subhereditary radicals; (ii) each non-zero left ideal of A contains a left ideal isomorphic (as a ring) to A. A ring A is called left strongly prime if each non-zero ideal I of A contains a finite subset F such thatfx 2 A j xf =0g = 0 and the left strongly prime radical is defined as the upper radical determined by the class of all left strongly prime rings. Right strongly prime rings and the right strongly prime radical are defined by the dual conditions. In [69], Sands proved that each radical which is left or right hereditary and left or right strong, satisfy all these properties. In [58] it is mentioned that left (right) subhereditary radicals are dual to left (right) strong radicals. In a similar manner left (right) hereditary radicals are dual to left (right) stable radicals. A radical fl is said to be left (right) stable if its semisimple class is left (right) hereditary. In [58], Puczyψlowski and Zand posed the following problems. (i) (Sand's question) Is the Brown McCoy radical left subhereditary? (ii) Is the left (right) strongly prime radical left subhereditary? (iii) Does a radical, which is left or right subhereditary and left or right stable, satisfy all these conditions? Next, we shall consider recent results on those problems. We set r A (B) =fa 2 A j Ba =0; B Ag and l A (B) =fa 2 A j ab =0; B Ag: Let B be a left ideal of A and let b 2 B. We denote by R b the endomorphism of the module A B given by (x)r b = xb for all x 2 B. Clearly the map ff : B! End( A B)given by b ff = R b is a homomorphism of rings and Ker(ff) =r B (B):

13 Some issues in the theory of supernilpotent radicals 13 Note that for any b; x 2 B and f 2 End( A B), (x)(r b f)=((x)r b )f =(xb)f = x(b)f =(x)r (b)f : Hence R b f = R (b) f. This in particular implies that B ff / r End( A B). The following lemma will be used several times. Lemma If L is an essential left (respectively, right) ideal of A and L is a semiprime ring, then l A (L) =0(respectively, r A (L) =0). Proposition 5.12 (Beidar, Ke and Puczyψlowski [11]). In the above notation, if r B (B) =0, then B ff ο = B and B ff is an essential right ideal of every subring S of End( A B) containing B ff. Given a ring A and a non-empty setx we shall denote by MX c (A) (respectively, MX r (A)) the ring of matrices over A indexed by elements from X, whichhave only finitely many non-zero columns (respectively, rows). Moreover we set M f X (A) = MX c (A) M X r (A) (obviously if X is finite, then M X c (A) =M X r (A) =M f X (A) isthe full ring of matrices over A indexed by elements from X). Observe thatm f X (A) is aright ideal of MX c (A) and a left ideal of M X r (A). For a given commutative domain K with unity and a given set X we denote by KhXi (respectively, K Λ hxi) the free K-algebra without unity (respectively, the free K-algebra with unity) in indeterminates from the set X. Proposition 5.13 (Beidar, Ke and Puczyψlowski [11]). For every non-empty set X and arbitrary commutative domain K with unity, KhXi is isomorphic to an essential right ideal of M c X (KhXi) and to an essential left ideal of M r X (KhXi). Remark The embedding of KhXi into MX c (KhXi) described in Proposition 5.13 can be presented directly as follows: given a = a 1 x a n x n 2 KhXi, where a i 2 K Λ hxi and x i 2 X, we map a onto the matrix, whose xx i -entry for x 2 X and 1» i» n, is equal to xa i and all the other are equal to zero. Theorem 5.15 (Beidar, Ke and Puczyψlowski [11]). Given a radical fl the following are equivalent: (i) fl is right subhereditary and for every infinite set X, the ring MX c (ZhXi) is fl-radical. (ii) fl is left subhereditary and for every infinite set X the ring MX r (Zhxi) is flradical. (iii) fl is hereditary and all fl-semisimple rings satisfy a common polynomial identity. The following corollary give a negative answer to the question (i). Corollary The Brown McCoy radical G is neither left nor right subhereditary. The following corollary gives a negative answer to question (ii). Corollary 5.17 (Beidar, Ke and Puczyψlowski [11; 12]). (i) The left strongly prime radical is not left subhereditary. (ii) The right (left) strongly prime radical is not left (right) subhereditary.

14 14 S. Tumurbat Proposition 5.18 (Beidar, Ke and Puczyψlowski [11]). Suppose that fl is a radical such that (i) if fl contains a non-zero torsion free ring, then all fl-semisimple rings are reduced; (ii) if fl contains for a prime p a non-zero Z p -algebra, then all fl-semisimple Z p - algebras are reduced. Then fl is left subhereditary if and only if it is right subhereditary. The following theorem gives an affirmative answer to question (iii). Theorem 5.19 (Beidar, Ke and Puczyψlowski [11]). Every left or right subhereditary and left or right stable radical satisfies all these conditions. In [58], Puczyψlowski and Zand also suggested to characterize a left subhereditary upper radical UM by conditions imposed on the class M. Next, we shall address this issue. Let us consider the following two conditions a class M of rings may satisfy (cf. [82]). (a) If L/ l A and L 2M, then A has a non-zero homomorphic image in M. (b) If L/ l A, L 2SUMand (L + I)=I =2Mnf0g for every ideal I of A, then 0 6= A=K 2Mwith an appropriate ideal K of A. Theorem 5.20 (Tumurbat and Wiegandt [82]). (i) Let M be a regular class of rings. The upper radical UM is left subhereditary if and only if M satisfies conditions (a) and (b). (ii) A hypernilpotent radical fl = UM is left subhereditary if and only if the semisimple class Sfl satisfies condition (a) for every semiprime ring A. Corollary A special radical fl is left subhereditary if and only if the class Sfl P satisfies conditions (a) and (b). Let us consider the subclass! = fa j L/ l A and L 2SUMimply (L + I)=I 2Mnf0g for every I/Ag which consists of rings A satisfying the assumption of condition (b). In general, it may happen that! is not empty. Corollary Let M be a regular class of rings satisfying condition (a). If the class! is empty, then fl = UM is left subhereditary. Corollary A radical fl is left subhereditary if and only if its semisimple class satisfies condition (a). The following lemmas will be used several times later on. Lemma (i) In a prime ring A every non-zero left ideal L is essential. (ii) If L/ l A and A is a semiprime ring, then L I/ l I for every 0 6= I/A.

15 Some issues in the theory of supernilpotent radicals 15 Lemma (i) If a non-zero left ideal L of a ring A has a unity element e, then L = Ae is a homomorphic image of A and e is the unity element of Ae. (ii) If a non-zero left ideal L of a ring A is right strongly prime, then A has a non-zero right strongly prime homomorphic image. Corollary The class % of all right strongly prime rings cannot satisfy condition (b). Proposition 5.27 (Tumurbat and Wiegandt [82]). Conditions (a) and (b) are logically independent. In fact, (i) the class C of all simple rings with unity element satisfies condition (a) and cannot satisfy condition (b); (ii) the class M of all reduced rings satisfies condition (b), but cannot satisfy condition (a). Now, we shall focus our interest on the Brown McCoy radical G. The following Proposition 5.28 and Lemma 5.29 will be used in the next results. Proposition 5.28 (Tumurbat and Wiegandt [82]). If a subdirectly irreducible prime ring A contains a Brown McCoy semisimple left ideal L 6= 0, then A is a simple ring with unity element. Lemma Let A be a subdirectly irreducible ring with nilpotent heart. If 0 6= L/ l A, thenfi(l) 6= 0. Theorem 5.30 (Tumurbat and Wiegandt [82]). The following two conditions are equivalent. (i) A is a simple ring with unity element; (ii) A is a subdirectly irreducible ring, and possesses a Brown McCoy semisimple essential left ideal. An ideal M of a ring A is said to be representable, ifm = T K, where each K is an ideal of A such thata=k is a simple ring with unity element. Theorem 5.31 (Tumurbat and Wiegandt [82]). If a ring A 6= 0 has a Brown McCoy P semisimple essential left ideal L, then A is Behrens semisimple. If A = A=I ν is a subdirect decomposition of A into subdirectly irreducible rings A=I ν ν2# with non-zero idempotents in the hearts and each L I ν is a representable ideal in L, thena is Brown McCoy semisimple. A ring A is called strongly ff-semisimple with respect to a semisimple class ff, if every homomorphic image of A is in the class ff. As is known from Andrunakievich [2], a ring A is strongly Brown McCoy semisimple if and only if each of its ideals is representable. Corollary (i) If 0 6= L/ l A and L is strongly Brown McCoy semisimple, then A is Brown McCoy semisimple. (ii) If 0 6= L/ l A and L is Brown McCoy semisimple, A = P ν2 A=I ν is a subdirect decomposition as given in Theorem 5.31 such that for some ν 2, L I ν is a

16 16 S. Tumurbat representable ideal in L, thenthere exists an ideal I of A such that A=I is a simple ring with unity element. Proposition 5.33 (Tumurbat and Wiegandt [82]). If L/ l A, G(L) =0and A has dcc on principal ideals, then G(A) =0. Proposition 5.34 (Tumurbat and Wiegandt [82]). If 0 6= L/ l A and G(A) =A and A has dcc on principal ideals, then G(L) 6= 0. Let A = X subdirect (A j 2 Λ) be a subdirect sum of rings A. McCoy [44] calls the ring A a special subdirect sum, if (0;:::;A ; 0 :::) A for each index 2 Λ, that is, A contains the discrete direct sum of the rings A, 2 Λ. [44, Theorem 6] says that a Brown McCoy semisimple ring A is a special subdirect sum of simple rings with unity if and only if every ideal of A contains a minimal ideal (which is a simple ring) with unity element. Such Brown McCoy semisimple rings are said to be special. A Brown McCoy semisimple ring A is called completely non-special, ifa contains no minimal ideals with unity element. For equivalent definitions we refer to [74, Definition 6]. Theorem 5.35 (Tumurbat and Wiegandt [82]). If A is a non-zero Brown McCoy radical ring and L/ l A such that G(L) =0, then every accessible subring K of L is completely non-special and K does not contain strongly Brown McCoy semisimple ideals. A ring A is prime essential, if A is semiprime and every prime ideal of A has a non-zero intersection with each non-zero ideal of A. This notion is due to L. H. Rowen [29; 26]. Theorem 5.36 (Tumurbat and Wiegandt [82]). There exists a prime essential Brown McCoy radical ring B having an essential left ideal which is a prime essential ring and completely non-special Brown McCoy semisimple ring, and this left ideal does not contain strongly Brown McCoy semisimple ideals. Remark For any cardinality! χ 0, there exists a ring B in Theorem 5.36, such thatjbj =!. Theorem 5.38 (Tumurbat and Wiegandt [82]). Let A be a special subdirect sum of subdirectly irreducible prime rings A, 2 Λ. If 0 6= K/L/ l A and K is Brown McCoy semisimple, then at least one A is a simple ring with unity element, whence A =2G. Compiling and using statements of the above results, we arrive at Corollary If a Brown McCoy semisimple ring L is embeddable as an essential left ideal into a ring A, thena is Behrens semisimple. If, in addition, A is a Brown McCoy radical ring, then (i) none of the subdirect decompositions

17 Some issues in the theory of supernilpotent radicals 17 A = X subdirect (A=I ν j A=I ν is subdirectly irreducible and A=Iν 2 SB) is special, (ii) every accessible subring K of L is completely non-special and K does not contain strongly Brown McCoy semisimple ideals, (iii) (L + I ν )=I ν is not Brown McCoy semisimple for each index ν, (iv) (L + J)=J is not a simple ring with unity elements for all ideals J of A. It may happen that both A and L are prime essential rings. Next, we shall consider some results on left subhereditary and hypernilpotent radical. Proposition 5.40 (Mendes and Tumurbat [47]). If fl is a left subhereditary radical and A is any semiprime ring, then every fl-semisimple left ideal L of A can be extended to a unique largest ideal I of A such that fl(a) I =0. Proposition 5.41 (Mendes and Tumurbat [47]). Let fl be a hypernilpotent radical. If in every semiprime ring A each fl-semisimple left ideal L can be extended to a unique largest ideal I of A such that fl(a) I =0, then fl is left subhereditary. Corollary Let fl be a left subhereditary radical. If a semiprime ring A has a non-zero-fl-semisimple left ideal, then either fl(a) =0or A is not a subdirectly irreducible ring. Theorem 5.43 (Mendes and Tumurbat [47]). Let fl be a hypernilpotent radical. Then the following are equivalent: (i) fl is left subhereditary. (ii) If A is a prime ring, L/ l A and fl(a)l 6= 0, then fl(l) 6= 0. (iii) In every semiprime ring A each fl-semisimple left ideal L of A can be extended to a unique largest ideal I of A such that fl(a) I =0. Theorem 5.44 (Mendes and Tumurbat [47]). For a special class M, the following conditions are equivalent: (i) fl = UM is left subhereditary. (ii) If A is a semiprime P ring with fl(a) 6= 0and L/ l A, thenfl(l)l 6= 0. (iii) Let B = (B 2 fl P j there exists 0 6= L / l B with L =r L (L ) 2 subdirect M). If L/ l B and L 2 Sfl, thenb 2 Sfl. A class M of rings is called a left special (left weakly special) if the following conditions are satisfied: (i) M consists of prime (semiprime) rings, respectively. (ii) M is hereditary. (iii) If A is a semiprime ring, 0 6= L/ l A and L 2M,thenA 2M. Next, we shall consider some results on the lattice of supernilpotent left subhereditary radicals.

18 18 S. Tumurbat Theorem 5.45 (Mendes and Tumurbat [47]). (i) A hypernilpotent radical fl is left subhereditary if and only if Sfl is left weakly special class of rings. (ii) A hypernilpotent radical fl is a left subhereditary and special if and only if Sfl is a left weakly special and fl = U(Sfl P ). A radical fl is said to be left special (left weakly special) iffl is the upper radical of a left special (a left weakly special) class. Lemma Let ffl i j i 2 Λg be a family of left subhereditary weakly special radicals. Then L([fl i ), the lower radical class determined by [fl i, is also a left subhereditary and weakly special radical class. Theorem 5.47 (Mendes and Tumurbat [47]). A radical class fl contains a largest left subhereditary special radical class if and only if it contains the class Z of all zero rings. Corollary A radical fl contains a largest left subhereditary radical class if and only if fl contains a largest weakly special radical class. Theorem 5.49 (Mendes and Tumurbat [47]). The following conditions are equivalent for a class M of rings: (i) UM is a radical class containing a left subhereditary special radical class. (ii) UM is a radical class containing a left subhereditary weakly special radical class. (iii) (a) Every 0 6= A 2Mhas a non-zero image in SUM. (b) SUM E where E denotes the class of all semiprime rings. Corollary 5.50 (see [46]). The following conditions are equivalent for an arbitrary class M of rings. (i) UM is a radical class containing a left subhereditary special radical class. (ii) UM is a radical class containing a left subhereditary weakly special radical class. (iii) UM is a radical class containing a special left strong radical class. (iv) UM is a radical class containing a weakly special left strong radical class. (v) UM is a radical class containing a left special radical class. (vi) UM is a radical class containing a left special radical class. (vii) UM is a hypernilpotent radical class. For a hereditary class M of semiprime rings, let us define M K = fa j A has an essential left ideal in SUMg: Theorem 5.51 (Mendes and Tumurbat [47]). Let M be a hereditary class of semiprime rings. Then the following statements are equivalent: (i) UM is a left subhereditary radical class. (ii) UM = UM K. (iii) UM M K =0. (iv) M K SUM. (v) Every 0 6= A 2 M K has some non-zero image in SUM.

19 Some issues in the theory of supernilpotent radicals 19 (vi) UM has the intersection property relative to M K ; that is for any ring A, the radical UM(A) = fi j A=I 2M K g. Clearly, a left hereditary radical is left subhereditary. Now, we shall focus our interest on the semisimple class of left hereditary radicals. A left ideal L of a ring A is called w-essential if w/ L/ l A and I L w imply 6= I =0,foranyidealI of A. We denoteitby L/ w l A and if L/A,thenby L/ w A. Proposition Let M be a regular class of rings. The upper radical fl = UM is hereditary, whenever M satisfies the following condition: If I/ w A, I=W 2Mand W 3 =0, then A has a non-zero homomorphic image in M. Theorem 5.53 (Sands and Tumurbat [72]). Let M be a hereditary class of rings which satisfies the following condition: If either LW = 0 or (L=W ) 2 = 0 for L/ w l A and L=W 2 M then A has a non-zero homomorphic image in M. Then if fl = UM is hereditary, then fl is left hereditary. The following Lemma is needed in the proof of Theorems 5.55 and Lemma Let fl be a left hereditary radical. If L/ w l A 2 Sfl. A and L=W 2 Sfl, then Theorem 5.55 (Sands and Tumurbat [72]). Let M be a hereditary class of rings. Then the upper radical UM is left hereditary if and only if M satisfies the following condition: Let L/ w l A and L=W 2 M. If either W 3 = 0 or (L=W ) 2 = 0, then A has a non-zero homomorphic image in M. Mendes and Wiegandt [48] asked for a characterization of the semisimple classes of left hereditary radicals. The next theorem and corollary give such characterizations. Theorem 5.56 (Sands and Tumurbat [72]). Let fl be aradical. Then the following are equivalent: (i) fl is left hereditary. (ii) Let L/ w l A and L=W 2 Sfl. If either LW =0or (L=W ) 2 =0, then A 2 Sfl. (iii) Let L/ w l A and W = fl(l). If either Lfl(L) = 0 or (L=fl(L)) 2 = 0, then A 2 Sfl. Corollary (i) Aradical fl is left hereditary if and only if given a ring A and left ideal L of A, iffori/a, I L fl(l) imply I =0, then A 2 Sfl. (ii) Aradical fl is right hereditary if and only if given a ring A and right ideal R of A, if for I/A, I R fl(r) imply I =0, then A 2 Sfl. Lemma Let S be a subring of a ring A and S=W be a non-zero prime (semiprime) ring. If an ideal I of A is a maximal with respect to I S W, then A=I is a prime (semiprime) ring, respectively.

20 20 S. Tumurbat Proof. Suppose A=I is not prime (semiprime), respectively. Then there are ideals K 1 and K 2 of A, such that I ρ K 1 and I ρ K 2 and also K 1 K 2 I (ideal K of A such thatk 2 I and K 6 I) respectively. Since I is a maximal ideal of A with respect to I S W,wehave I 1 = K 1 S 6 W and I 2 = K 2 S 6 W (J = K S 6 W ), respectively. Clearly I 1 /S and I 2 /S (J /S). Therefore and respectively, acontradiction. (I 1 + W )=W 6= 06= (I 2 + W )=W (I 1 + W )=W (I 2 + W )=W = 0((K + W ) 2 W ); A radical fl is said to be strongly hereditary if a subring S of a ring A 2 fl then S 2 fl. Proposition A radical fl is strongly hereditary if and only if, given a ring A and a subring S of A, if for I/A, I S fl(s) imply I =0, then A 2 Sfl. Proof. Let S be a subring of a ring A, and if I S fl(s) fori/a,theni =0. Suppose fl(a) 6= 0. Since fl(a) S/S and fl is strongly hereditary fl(a) S is a fl-radical ring and fl(a) S fl(s). By assumption fl(a) = 0, a contradiction. (: Let S be a subring of A 2 fl. Suppose S =2 fl. Hence fl(s) 6= S. Let I be a maximal ideal of A with respect to I S fl(s). Put A = A=I and S =(S + I)=I. Clearly, S is a subring of A and fl(s) =fl((s + I)=I) ο = fl(s=s I) =fl(s)=s I: Hence if J/A and J S fl(s), then J =0. Thus A 2 Sfl fl = 0, and A = I, but S I S fl(s) and S = fl(s), a contradiction. Λ Theorem (i) Let fl be a strongly (left, right) hereditary hypernilpotent radical. Then the U(Sfl P ) is strongly (left, right) hereditary special radical, respectively. (ii) Let fl be aradical, fl is a strongly (left, right) hereditary special radical if and only if given a prime ring and subring (left ideal, right ideal, respectively) L if for I/A, I L fl(l) imply I =0,thenA2Sfl, andfl = U(Sfl P ). Proof. (i) Let S be a subring (left ideal, right ideal) of a ring A whichisinu(sfl P ). Suppose S =2U(Sfl P ). Then S has a non-zero homomorphic image S=W in Sfl P. Let I be a maximal ideal of A with respect to I S W. By Lemma 5.58, A=I is a prime ring. Since A=I 2U(Sfl P ), we have 06= fl(a=i) =K=I, for K/A. Since I is a maximal ideal with respect to S I W, S K 6 W,andS K/S. Therefore we have 0 6= (S K)=(S K W ) ο = ((S K)+W )=W / S=W 2 Sfl P and (S K)=(S K W ) 2 Sfl P. Moreover, ((S K) +I)=I is a subring (left ideal, right ideal, respectively, of fl(a=i). Since fl is a strongly (left, right, Λ

21 Some issues in the theory of supernilpotent radicals 21 respectively) hereditary ((S K) +I)=I 2 fl and also (S K)=(S K I) 2 fl. Since S I W,wehave S K I S K W. Therefore, (S K)=(S K W ) is a homomorphic image of (S K)=(S K I). Thus (S K)=(S K W ) 2 fl, a contradiction. Since fl is hereditary radical U(Sfl P ) is a special radical. (ii) ). Since fl is a special radical, fl = U(Sfl P). The assumptions imply from Proposition 5.59 (Corollary 5.57 (i) and (ii), respectively). (: Let S be a subring (left ideal, right ideal) of A 2 fl and S =2 fl. Since fl = U(Sfl P ), S has a non-zero homomorphic image S=W in Sfl P. We can choose a prime ideal I of A such thati is maximal with respect to S I W. Put S =(S + I)=I, W =(W + I)=I and A = A=I. Since S I W,wehave fl(s) W. Let J = J=I be an ideal of A, such that J S fl(s) W. Then (J + I) (S + I) fl(s + I) W + I, and it follows that J S W and J = I. So J =0. Since A is a prime ring, by assumption A 2 Sfl fl =0,acontradiction. Λ Remarks (i) The Brown McCoy radical G is a special radical which is not left (right) hereditary. Moreover G is not principally left (right) hereditary. (ii) The Behrens radical B and the Jacobson radicals are left and right hereditary special radicals, but not strongly hereditary. (iii) Köthe's nil radical N, the locally nilpotent radical L and the Baer radical fi are strongly hereditary special radicals. Rossa and Tangeman [63] called a relation ff on the class of rings an H-relation, if ff satisfies the following properties: (i) BffA implies B is a subring of A, (ii) if BffA and f is a homomorphism of A, thenfbfffa, (iii) if BffA and I/A,then(B I)ffI. We shall assume that the H-relation ff satisfies also the following additional condition. (iv) If f is a homomorphism of A and fbfffa, then also BffA. Examples of such H-relations are ideal of", left ideal of", bi-ideal of", and subring of", beside many other H-relations (see [63], Propositions 1, 2 and 3). A subclass M of rings is said to be ff-hereditary, ifbffa 2Mimplies B 2M. In the special case ff = subring of" we speak of a strongly hereditary class M. We recall Theorem 4 of [63]. Proposition Let ff be an H-relation. If M is a homomorphically closed and ff-hereditary class of rings, then the lower radical L(M), generated by M is also ff-hereditary. Theorem Let M be a strongly (left, right) hereditary class of rings. The lower special radical L sp (M) and the lower hypernilpotent radical L h (M), generated by M are strongly (left, right, respectively) hereditary radicals. Proof. Clearly L h (M) =L(fi [M)=L(fi [LM). Put L 1 = fa j any subring of A is in L h (M)g;

22 22 S. Tumurbat and L 2 = fa j any left ideal of A is in L h (M)g: Clearly L 1 is a strongly hereditary class of rings and fi L 1 L 2. We claim that L 2 is a left hereditary class. Let A 2L 2 and K/ l L/ l A. Since LK / l A,wehave LK 2L h (M). But LK / K and (K=LK) 2 =0. Since fi L 2, K 2L h (M). Therefore L 1 and L 2 are strongly hereditary and left hereditary classes, respectively. Now weshallshow that L 1 and L 2 are homomorphically closed classes. Let A 2 L 1 and A 2L 2. We consider a homomorphic image A=I of A. Let S=I beasubring (left ideal) of A=I. Then S is a subring of A (left ideal) of A, respectively. Since A 2 L 1 and A 2 L 2, S is in L 1 and L 2, respectively. Therefore S 2L h (M), and so S=I 2L h (M). So L 1 and L 2 are homomorphically closed classes of rings. Hence M L 1 and M L 2. By Proposition 5.62, L h (M) =L(L 1 )andl h (M) =L(L 2 ) are strongly hereditary and left hereditary, respectively. For the right hereditary case of M, the proof is similar to L 2. Now, by Theorem 5.60 (i) L sp M is a strongly (left, right) hereditary radical respectively. Λ Many authors gave examples of supernilpotent, but not special radicals and also left hereditary supernilpotent, but not special radicals (see Example 3.12 and [61; 87; 16; 26]). Now we will give examples of left hereditary supernilpotent but not special radicals. Moreover, we will give examples of strongly hereditary supernilpotent but not special radicals. Theorem 5.64 (Gardner and Stewart [29]). Let fl be aradical class and % a class of fl-semisimple rings such that (a) every semiprime homomorphic image of a ring in % is prime and (b) there is a prime essential ring E such that every prime homomorphic image of E is in %. Then L(fl [ %) U(fEg) andnoradical class in the interval [L(fl [ %); U(fEg)] is special. Lemma 5.65 (Rjabukhin [29]). Let S be a linearly ordered set with no greatest element, with least element e and which is such that every interval [x; y], x <y, has cardinality k. Then S is a semi-group with multiplication defined by xy = maxfx; yg. Let A be a non-zero semiprime ring. (i) The semigroup ring A(S) is a subdirect product of copies of A. (ii) A(S) is semiprime. (iii) If I is a prime ideal of A(S), then P = fa 2 A j ae 2 Ig is prime ideal of A and A(S)=I ο = A=P. Put F f = fall finite fieldsg, and F = fall fieldsg and also fl f = U(F f ). Corollary For any left hereditary subclass S f of fl f and non-empty subclass F s of F (i) the lower radical L(S f [ F s ) is non-special and if Z 0 2 S f, then L(S f [ F s ) is left hereditary, where Z 0 is the zero-ring on the integers;