ADDITIVE GROUPS OF SELF-INJECTIVE RINGS

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1 SOOCHOW JOURNAL OF MATHEMATICS Volume 33, No. 4, pp , October 2007 ADDITIVE GROUPS OF SELF-INJECTIVE RINGS BY SHALOM FEIGELSTOCK Abstract. The additive groups of left self-injective rings, and rings belonging to classes properly containing the left self-injective rings, are studied. A complete description is given of the torsion groups and torsion free groups which are additive groups of left self-injective rings. 1. Introduction In this note rings are assumed to have an identity element, and modules are left modules. The additive group of a ring R will be denoted R +. Recall that a ring R is a left CS-ring if every left ideal of R is essentially contained in an R-module direct summand Re of R, with e an idempotent in R. If R is a left CS-ring, and every left ideal in R isomorphic to an R-module direct summand of R is itself a direct summand of R, then R is a left continuous ring. It is well known that every left self-injective ring is a left continuous ring. Recall that a ring R is von Neumann regular if for every element a R there exists an element b R such that aba = a. All groups in this note are abelian, with addition the group operation. An element a in a group G is a torsion element if there exists a positive integer n such that na = 0. If every element of a group G is a torsion element then G is said to be a torsion group. If 0 is the only torsion element in the group G then G is a torsion free group. A group G for which there exists a positive integer n satisfying nx = 0 for all x G is called a bounded group. Lemma 1. Let G be a torsion free group. The following are equivalent: Received November 16, 2005; revised January 18, AMS Subject Classification. 20K99. Key words. 641

2 642 SHALOM FEIGELSTOCK (1) G is the additive group of a left self-injective ring. (2) G is the additive group of a left continuous ring. (3) G is divisible. Proof. Clearly (1) implies (2). Let R be a left continuous ring satisfying R + = G. For every positive integer n, the left ideal nr is isomorphic to R as an R-module. Therefore nr is a direct summand of R which clearly implies that nr = R, so (2) implies (3). A divisible torsion free group is the additive group of a field, hence (3) implies (1). Theorem 2. Let G be a torsion group. The following are equivalent: (1) G is the additive group of a left self-injective ring. (2) G is the additive group of a left continuous ring. (3) G is the additive group of a left CS-ring. (4) G is the additive group of a ring with identity. (5) G is bounded. Proof. The implications (1) (2) (3) (4) are obvious. Let R be a ring with identity satisfying R + = G. Since 1 x = 0 for all x R it follows that G is bounded, so (4) (5). Let G be a bounded group. There are only finitely many primes p for which the p-component G p of G is non-zero. Direct products of left self-injective rings are left self-injective, so it may be assumed that G is a p-group. There exists a positive integer n such that G = n Z(p k ) with α k a cardinal for 1 k n. k=1 a k Again employing the fact that direct products of left self-injective rings are left self injective, it may be assumed that G = Z(p k ) with k a positive integer, and α a α cardinal. If α is finite then the product of α copies of Z/p k Z is a left self-injective ring with additive ring isomorphic to G. If α is infinite let X = {X i : i α} be a set of variables. Put R = Z/p k Z[X] and let S be the localization of R at the ideal pr. If A is a (left) ideal in S and ϕ : A S is an S-homomorphism, then A = p t S for some 0 t < k and ϕ(p t ) = p m u with u a unit in S and m t. The map ψ : S S defined by ψ(s) = sp m t u for all s S is an S-homomorphism

3 ADDITIVE GROUPS OF SELF-INJECTIVE RINGS 643 extending ϕ, so S is a left self-injective ring with additive group isomorphic to G. The following result of Faith reduces the problem of determining the additive groups of left self-injective rings to two cases: regular rings, and rings which do not contain non-zero regular ideals. Here regular means von Neumann regular. Proposition 3. If R is left self-injective then R is a ring direct sum R = S T with S a regular ring, and T a ring with no non-zero regular ideals. Proof. [1]. The following almost complete description of the additive groups of regular groups was obtained by Fuchs [4]. Proposition 4. Let R be a regular ring. Then the additive group of R is R + = D C with D a divisible torsion free group, C a reduced group satisfying t(r) = R p C R p with R p an elementary p-group for each prime p, and p p C/t(R) is a divisible torsion free group. Here t(r) is the torsion part of R +. Proof. [5, Theorem 124.1], or [3, Theorem 1.2.2]. The following is a result of Goodearl, [6, Proposition 3]. Proposition 5. Let R be a regular left self-injective ring. Then R is an indecomposable ring if and only if R is a prime ring. Proof. [7, Proposition 9.6]. Proposition 6. Let R be a semiprime ring. An idempotent e R is central if and only if er(1 e) = 0. Proof. [7, Lemma 3.1]. The following result is a generalization of Proposition 5 for left CS-rings. The proof will be given even though it is very similar to the proof of Proposition 5 given in [7, Proposition 9.6]. Theorem 7. Let R be a regular left CS-ring. Then R is an indecomposable ring if and only if R is a prime ring.

4 644 SHALOM FEIGELSTOCK Proof. Let I,J be non-zero ideals in R satisfying IJ = 0, and let K = {a R : aj = 0}. There exists an idempotent e R such that K is essentially contained in Re. For any x R one has Kx K Re, so Kx(1 e) = 0. The map ϕ : Re Rex(1 e) via right multiplication by x(1 e), is an R- epimorphism with K ker ϕ, so Rex(1 e) is an epimorphic image of Re/K. Since Re/K is a singular R-module it follows that Rex(1 e) is singular. There are no non-zero singular left ideals in a regular ring, so Rex(1 e) = 0 which implies that er(1 e) = 0. Proposition 6 yields that e is a central idempotent in R, and so Re is a non-trivial ring direct summand of R, a contradiction. The non torsion-free groups which are additive groups of prime rings are completely described in the following Proposition. Proposition 8. A non torsion free group is the additive group of a prime ring if and only if it is an elementary p-group, p a prime. Proof. [2, Theorem 4.1.1]. Corollary 9. Let G be an abelian group. The following are equivalent: (1) G is the additive group of an indecomposable regular left self-injective ring. (2) G is the additive group of an indecomposable regular left CS-ring. (3) Either G is a divisible torsion free group or G is an elementary p-group with p a prime. Proof. Clearly (1) implies (2). Suppose that G is the additive group of an indecomposable regular left CSring. Then R is a prime ring by Theorem 7, so if G is not torsion free then G is an elementary p-group by Proposition 8. If G is torsion free then G is divisible by Proposition 4. If G is either a divisible torsion free group or an elementary p-group then G is the additive group of a field, so (3) implies (1). Corollary 10. A torsion group G is the additive group of a regular left self-injective ring if and only ng = 0 for some square free positive integer n. Proof. Let R be a regular left self-injective ring with R + = G. Then G is bounded by Theorem 2. If 0 a R has square additive order then a is clearly not regular.

5 ADDITIVE GROUPS OF SELF-INJECTIVE RINGS 645 Conversely, if ng = 0 for n a square free integer, then G is the additive group of the direct product of finitely many fields. The author is indebted to Professor Manfred Dugas for his help. References [1] C. Faith, The maximal regular ideal of self-injective and continuous rings splits off, Arch. der Math., 44(1985), [2] S. Feigelstock, Additive Groups of Rings, Research Notes in Mathematics 83, Pitman, Boston-London, [3] S. Feigelstock, Additive Groups of Rings II, Research Notes in Mathematics 169, Longman, Harlow, [4] L. Fuchs, Ringe und ihre additive gruppe, Publ. Math. Debrecen, 4(1956), [5] L. Fuchs, Infinite Abelian Groups II, Academic Press, New York-London, [6] K. R. Goodearl, Prime ideals in regular self-injective rings, Canad. J. Math., 25(1973), [7] K. R. Goodearl, Von Neumann Regular Rings, Pitman, London, Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel. feigel@math.biu.ac.il