Shalom Feigelstock

Rings whose additive endomorphisms aren-multiplicative, II

A ringR is called anAE n -ring,n≥2 a positive integer, if every endomorphism ϕ of additive group ofR satisfies ϕ(a 1 a 2...a n )=ϕ(a 1)ϕ(a 2)...ϕ(a n ) for alla 1,...,a n eR. Several results concerning the structure ofAE n -rings are obtained in this note, including an (incomplete) description ofAE n -ringsR satisfyingR t R n−1 ≠0, whereR t is the torsion ideal inR.

Abelian groups mapping onto their endomorphism rings

An abelian group G is said to be an E-group if it is the additive group of an E-ring. It is known that G is an E-group if and only if there exists a left E(G)-module isomorphism from G to its endomorphism ring E(G). Groups which are isomorphic to the additive group of their endomorphism rings are called weak E-groups. The purpose of this article is to consider the apparently yet weaker condition that there be a homorphism from G onto the additive group of E(G). Groups satisfying this condition are called EE-groups. The properties of EE-groups are studied and it is shown that they are very similar to E-groups. In fact, it is shown that every EE-group of finite torsion-free rank is a weak E-group, and that for various prominent classes of groups the concepts of EE-group and E-group coincide.

Additive groups of rings whose subrings are ideals

An Abelian group G is called an SI -group if for every ring R with additive group R + = G , every subring S of R is an ideal in R . A complete description is given of the torsion SI -groups, and the completely decomposable torsion free SI -groups. Results are obtained in other cases as well.

The additive groups of local rings

All groups considered in this paper are abelian, with addition as the group operation. A ring R is said to be local if R is a ring with unity, and if R possesses a unique maximal ideal, i.e., the ideal of non-units in R. Necessary and sufficient conditions will be obtained for a torsion group G to be the additive group of a local ring. Necessary conditions will be given for a nontorsion free group to be the additive group of a local ring.

Self-Free Modules and E-Rings

Abstract We call an S -module M self-free with basis X if X is a subset of M and each function f : X → M extends to a unique endomorphism ϕ : M → M. Such structures were called minimally ∣X∣-free in Bankston and Schutt (Bankston P., Schutt, R. (1995). On minimally free algebras Can. J. Math. 37(5) : 963–978.). If |X|= 1, then M is an E-module over S. We show that if M is slender, then M is a direct sum of |X| many copies of an E-module, without restrictions on the cardinality of X. We also investigate additive groups of torsion-free rings of finite rank without zero-divisors. We find criteria under which these rings are E-rings. Also, we find conditions for abelian groups to be E-groups.

Some aspects of relative injectivity

If H and M are right R -modules, H is M -injective if every R -homonorphism N → H , N a right R -submodule of M , can be extended to an R -homonorphism from M to H . H is strongly M -injective if H is injective for inclusions whose cokernels are isomorphic to factor modules of M . For the case of abelian groups H and M , one settles the questions “when is H M -injective” and “when is H strongly M -injective”. The latter can be characterized in terms of the vanishing of Ext. Results for general module categories are also given.

Additive Groups of Local Near-Rings

ABSTRACT The object of this note is to obtain information about the additive groups of local near-rings. It will be shown that the additive group of a torsion local near-ring is a bounded p-group. The torsion Abelian groups which are additive groups of local near-rings will be described completely. A method will be given to construct groups of almost any prime power order, which are not additive groups of local near-rings.

Additive Groups of Commutative Rings

Abelian groups G, with addition the group operation, satisfying the property that every ring R with additive group R + = G is commutative, are studied. A complete description of these groups is obtained for torsion groups and completely decomposable torsion free groups.

A note on subdirectly irreducible rings

Let R be a commutative subdirectly irreducible ring, with minimal ideal M . It is shown that either R is a field, or M 2 = 0. A construction is given which yields commutative sub-directly irreducible rings possessing nonzero-divisors, and nonzero nilpotent elements either with a unity element, or without. Such a ring without unity has been constructed by Divinsky. The same technique enables the construction of subdirectly irreducible rings with mixed additive groups.

On the nilstufe of the direct sum of two groups

1. All groups in this paper will be assumed to be abelian groups with addition as the group operation. An associative operation on a group G satisfying both distributive laws will be called a multiplication on G. The multiplication gh=O for all g, h C G is called the zero multiplication, and can be defined on every group G. If G allows no other multiplication, then G is called a nil-group. The notion of a nilgroup was introduced by SZELE [2], and studied further in [1]. SZELE proved [1], [2] that a torsion group is nil iff it is divisible and that there are no nil mixed groups. In [3] SZELE generalized the concept of a nil-group as follows. Let n be a positive integer. If there exists a multiplication on G under which Gn#0, but G n+1-=0 under every multiplication, then G is said to have nilstufe n, denoted by v(G)=n, Clearly the nil-groups are the groups G with v(G)= 1. If v(G)>n for every positive integer n, then we will denote v(C)= co. The original intent of the research which led to this paper was to consider a group G = G 1 @G2, v(G1)=v(G2)=l , and to see what could be said about v(G). Szeles theorem assures us that neither G~ nor G 2 can be mixed. We further have that if G~ and G2 are both torsion groups then they are both divisible and G is therefore a divisible, torsion group, and hence nil by Szeles theorem. In theorem 1 we consider the case where G1 is a torsion group, and G 2 is torsion free. However, we consider there the more general situation, v(G1)=n, v(G2)=m, n, m arbitrary positive integers. In w 3 we study the case where both G1 and G 2 are torsion free and of rank 1. If either G1 or G 2 is not nil then its nilstufe is ~, ([1], p. 270). Therefore v(G~)=v(G2)=l is the most general case to consider when both G 1 and G 2 are torsion free groups of rank 1. Tbe case G~ and G2 torsion free and of arbitrary rank remains an open question. 2. THEOREM 1. Let G = Ga | G 2, G~ torsion ,v(Gt)=n, G 2 torsion free, v(G2)=m, then v(G) <= (n+l)(m+l)--l. PROOF. Consider the cartesian product G~XGj, i=1 or 2, j=l or 2. Every multiplication on G is a bilinear mapping of Gi X Gj into G and therefore factors through G~| Gj. If either i= 1 or j= 1 then G~| Gi is a torsion group, and g~gj C G~, gi ~ Gi, gj C G i. We therefore have: