PRIME AND SPECIAL IDEALS IN STRUCTURAL MATRIX RINGS OVER A RING WITHOUT UNITY

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1 J. Austral. Math. Soc. (Serie3 A) 45 (1988), PIME AND SPECIAL IDEALS IN STUCTUAL MATIX INGS OVE A ING WITHOUT UNITY L. VAN WYK (eceived 12 Setember 1986; revised 17 February 1987) Communicated by. Lidl Abstract A. D. Sands showed that there is a 1-1 corresondence between the rime ideals of an arbitrary associative ring and the comlete matrix ring M n () via P» M n (P). A structural matrix ring M(B, ) is the ring of all n x n matrices over with in the ositions where the n x n Boolean matrix B,B a quasi-order, has. The author characterized the secial ideals of M(B, '), in case ' has unity, for certain secial classes of rings. In this note results of Sands and the author are generalized to structural matrix rings over rings without unity. It turns out that, although the class of rime simle rings is not a secial class, Nagata's M-radical has the same form in structural matrix rings as the secial radicals studied by the author. 198 Mathematics subject classification (Amer. Math. Soc.): 16 A 21, 16 A Introduction We work entirely in the category of associative rings, and resuose a familiarity with the basic results of radical theory, most of which can be found in [6], and of secial classes of rings and secial radicals (see [1]). will be a generic symbol for a ring, and "ideal" will mean "two-sided ideal". B = [bij] will be a reflexive and transitive n x n Boolean matrix, that is, an n x n quasi-order, and E s t will be the n x n Boolean matrix with 1 in osition (a, t) and elsewhere. For ^ V C we set M(B, V) := {X = [xij] M B (V): bij - => x i3 = }, 1988 Australian Mathematical Society /88 SA

2 [2] Secial ideals in structural matrix rings 221 and we call M{B, ) a structural matrix ring. B determines and is determined by the binary relation <B on n := {1,2,..., n} defined by i <B j <> fyj = 1. This quasi-order relation naturally gives rise to the equivalence relation ~B on n defined by i ~B j : ** * <B j and j <B i. Let /? be the number of equivalence classes induced by ~B on n, and let z\, z%,..., z be their reresentatives. We use n^ to denote the cardinality of z^'s equivalence class, ft G /3, and yi,, V2,n, i Vn^,^ to denote its elements, with yi tli < j/t+i,^ for i = 1,2,..., n M 1. Consider the ring eimorhism / : M(B,) -> M n,,(jj) defined by / ([<: ]) = [d st ], where d st = c Vl iivt M ; 1 < s, t < n M. Then K := {X = [xij] M(B, ): x^ = if i ~B ^ and j ~B 2^} is the kernel of f^. Theorems 1 and 2 of [5] characterize the rime (res. rime maximal) ideals of the comlete matrix ring M n (/2) as the sets M n (P) corresonding to the rime (res. rime maximal) ideals P of, and in [7] the rime (res. maximal) ideals of M(B,'), ' a ring with unity, are characterized as the sets M(B,P) + K^ corresonding to the rime (res. maximal) ideals P of! and the ideals K^, \x e /?, of M(B, '). In fact, if C is a secial class of rings such that S C if and only if M n (5) C whenever S has unity, then roosition 2.6 of [7] states that the C-secial ideals of M(B, ') are the sets SA(B,P') + K, i corresonding to the C-secial ideals P' of ' and the ideals K^fi /?, of M(B,'). We generalize these results to structural matrix rings over a ring without unity and obtain similarities to the case of a ring with unity as far as rime ideals are concerned, but also a striking difference in the case of maximal ideals. Theorem 2.7 of [7] shows that for the uer radical class determined by a secial class C of rings satisfying the mentioned condition, the equality (M(B, )) = M(B, (.))+n M e/3 <ix holds for ev ery ring. The ideal fl^ K^ is called the antisymmetric radical of M(B,) in [7]. The M-radical of is the intersection of the rime ideals P of such that /P is a simle ring (see [4]), that is, the intersection of the rime maximal ideals of. We show that, although the class of rime simle rings is not a secial class (see Corollary 1 and Theorem 5 of [3]), the equality M(M(B,)) = M(B, M{)) + D Me/ 9 <it holds for the M-radical of a structural matrix ring. We first state the following two easily-roved generalizations of Lemmas 1.6 and 1.7 of [7]. LEMMA 1.1. Let V C. Then, V) + f K» = f (M(B, V) + /( ).

3 222 L. van Wyk [3] LEMMA 1.2. LetOeVC. Then f- 1 {M Ttl.{y)) = M(B,V) + K ll. We denote the set of rime ideals of by sec(i2). 2. Prime and secial ideals For all /x, G /? such that z M < B Z$, we set A M := {u G /?: z^ < B Z V and z,, < B Z«}- For a ring ' with unity, a standard argument using the matrix units shows that every ideal of M(B, ') has the form A e := {X = [xij\ G M(S,'): 2:^ (A^) if i ~ B z M, j ~ B z c and z M < B z e } for some set-inclusion reserving function : {A^: (i, f G /3 and z M < B Z$} {-4': A' is an ideal in '}, that is, a function such that A M $ C A^,, imlies ^(A M^) C POPOSITION 2.1. T/ie nme ideals of M(B, ) are the sets M(B, P) + <. corresonding to the rime ideals P of and the ideals /C^,/z /?, ofm(b,). POOF. Let /? > 1, otherwise the statement is just Theorem 1 of [5]. If P 6 sec(iz) and n e /?, then, by the same theorem, M n(j (P) G sec(m n>l (i2)), and so, by Lemma 1.2, M(B, P) + K^ G sec(m(b, i?)). Conversely, let P G sec(m(s, /?)), and imbed in a ring ' with unity such that is an ideal in /?'. It is trivially seen that M(B, ) is an ideal in M(B, '), which imlies that P is an ideal in M(B,') (see emark 2 (age 333) of [4]). Hence, by the remark receding this roosition, P = Ae for some set-inclusion reserving function : {A M^: /i, G /3 and z M < B z^} -+ {yl': A' is an ideal in '}. Let n, such that z^, < B z^, and set A^ := {a G : a = Xij for some * ~B ^M>i ~B ^{ and X = [x^] G P}. Then it follows from the definition of A$ that A^ = ^(A M^), and so (A^) is an ideal in. Furthermore, as mentioned for in the introduction, Z(M(B,)) = M(B,Z()) + fl/ie^ ^ ever y s P ecial radical k determined by a secial class C of rings such that S G C if and only if M n (5) G C whenever S has unity, and so in articular for the rime radical B (that is, Baer's lower radical), f Me/J <n Q B{M{B, )) C P. Therefore, ^M? = fi if n ^. However yl M/i 5^ i? for some fj, G /?, otherwise P = M (?,#). We assert now that {u G /9: A^^ ^ i?} = {^/}, for, if further A vv ^ i? for some rj ^ fi, then we consider the ideals A$> and Ae" of M(B,.) defined by otherwise,

4 [4] and Secial ideals in structural matrix rings 223 otherwise. Then Ao'A$" C P, but A$> < P and A$» < P, which contradicts the rimeness of P, and so our assertion is valid. It follows now from the form of Ae{= P) that P = M(B,A ll, i ) + Kf,, and so P contains the kernel of / M, which imlies that M n/4 (P) = / M (M(B,^4 MM ) + Ky) G sec(m n(i (i?)). This comletes the roof in the light of Theorem 1 of [5]. EXAMPLE 2.2. then the rime ideals of M(B, ) are If D " [ P and "i? i? 12 i? 1? i? P for P sec(ii). COOLLAY ie semi-rime ideals o/m(b,) are the sets Ae corresonding to the set-inclusion reserving functions maing all the h. vv 's onto semi-rime ideals of and all the A^ 'a, ^<r, onto. Let C be a secial class of rings such that T G C if and only if M n (T) C. Examles of such rings are the classes of simle rings with unity, strongly rime rings (see [2]) and, of course, rime rings. Note, however, that the latter condition is stronger than condition I in [1] and in Proosition 2.6 of [7]: I. If ' is a ring with unity, then ' e C if and only if M n {') C. Since every C-secial ideal is rime, we can now generalize Proosition 2.1. THEOEM 2.4. Let C be a secial class of rings such that T G C if and only ifm n (T) G C. The C-secial ideals ofm(b,) are the sets M{B,P) + Ky corresonding to the C-secial ideals P of and the ideals K^, fj. G /?, of M{B,).

5 224 L. van Wyk [5] POOF. See the roof of Proosition 2.6 of [7]. 3. Maximal and rime maximal ideals It follows from Theorem 2.4 that the maximal ideals M of M(B,) such that M(B, )/M has unity, are the sets M(B, N) + K^ corresonding to the maximal ideals N of such that /N has unity and the ideals K^,n G /J, of M(B,). Since a characterization of the maximal ideals of M n (), the "simlest" structural matrix ring, has not yet been obtained, one cannot exect the maximal ideals of M(B, ), in general, to have the form in Theorem 2.4. What is surrising, however, is that not even the antisymmetric radical Pl^e/} ^M f M(B, ) is, in general, contained in the maximal ideals of M(B, ), that is, the form of the ideals in Theorem 2.4 breaks down: EXAMPLE 3.1. Consider the ideal {,2,4,6} of Z 8 as the ring, and let N := {,4}. Then N is maximal, but not rime, in, and /N is without unity. If "1 1, \ B = [l lj' then [N is a maximal ideal in M(B, ). Note that this does not corresond to a setinclusion reserving function. The rime maximal ideals of a structural matrix ring still have the form in Theorem 2.4. THEOEM 3.2. The rime maximal ideals ofm(b, ) are the sets M(B, P)+ Kf, corresonding to the rime maximal ideals P of and the ideals K^, n /?, ofm(b,). POOF. We first show that every rime maximal ideal P of M(B,) has the asserted form. By Proosition 2.1, P M(B,P) + /C M for some P sec(i?), /i /?. If P is not maximal in, then M(B,P) + K^ is strictly contained in M(B,A) + Kfi ^ M(B,) for some roer ideal A of strictly containing P, which contradicts the maximality of M{B, P) + K fi. Hence, P is a rime maximal ideal of. Conversely, let P be a rime maximal ideal of, and let fi /J. Then, by Proosition 2.1, M{B,P) + K^ sec(m(b,}). Let M be an ideal of M(B,) strictly containing M{B,P) + <. Then f»(m) strictly contains U(M(B,P) + KJ = M nm (P), and so, by Theorem 2 of [5], / (>/) = M B(,(fl). Keeing in mind that M contains the kernel of f^, it follows from Lemma 1.2 that N = M(B, ), which comletes the roof.

6 [6] Secial ideals in structural matrix rings 225 We conclude this section by showing that the maximal ideals of M(B, ) which have the form M(B, P) + K^ for some (maximal, of course) ideal P of, are rime. POPOSITION 3.3. Let P be a maximal ideal of, and let fj. e P- Then M(B,P) + Kfi is maximal in M(B,) if and only if P is rime in. POOF. An easy argument using Theorem 4 of [5] and Theorem 3.2 yields the result. 4. Nagata's M-radical THEOEM 4.1. POOF. Let /z e P, and set Ly. := f\{m(b,p) + K^: P is a rime maximal ideal of }. Then, by Theorem 3.2, M(M(B, )) = P M /, M. But /C M, the kernel of f^, is contained in M, and so = Pi f l (P {/ M (M(S, P) + Kfi): P is a rime maximal ideal of I /ig/3 = P f~ x (P {Mn M (P) P is a rime maximal ideal of }j The assertion follows now from Lemmas 1.1 and 1.2. As in Corollary 2.9 of [7] we see that the M-radical carries over from to M(5, ) in the simlest ossible way, that is, M{M{B, )) = M(B, M{)) if and only if <B is an equivalence relation. eferences [1] V. A. Andrunakievich, 'adicals of associative rings I', Amer. Math. Soc. Transl. Ser. (2) 52 (1966), [2] N. J. Groenewald and G. A. P. Heyman, 'Certain classes of ideals in grou rings II', Comm. Algebra 9 (1981),

7 226 L. van Wyk [7] [3] G. A. P. Heyman and C. oos, 'Essential extensions in radical theory for rings', J. Austral. Math. Soc. 23 (1977), [4] M. Nagata, 'On the theory of radicals in a ring', J. Math. Soc. Jaan 3 (1951), [5] A. D. Sands, 'Prime ideals in matrix rings', Glasgow Math. Assoc. Proc. 2 (1965), [6] F. A. Szasz, adicals of rings, (John Wiley ii Sons, New York, 1981). [7] L. van Wyk, Secial radicals in structural matrix rings, Tech. reort 5/1986, Det. of Math., University of Stellenbosch. Deartment of Mathematics University of Stellenbosch 76 Stellenbosch South Africa