On Regular Near-Ring Modules

Abstract

In this paper we introduce the notion of a regular near-ring module by extending the usual elementwise definition of a regular near-ring to arbitrary near-ring modules. We characterize these modules in terms of certain restricted injectivity properties (Proposition 2.9). Using this characterization we deduce several characterizations of regular near-rings (Theorem 2.10). We also determine a characterization of strictly semisimple near-rings among near-rings with no nonzero nilpotent elements (Theorem 2.13). Throughout, R will denote a right near-ring with identity 1 such that x0 = 0 for all x ∈ R, and all modules (that is, near-ring modules) over R are left unital. The symbol RM will denote a left near-ring module M over R, and the term R-subgroup of M will mean a subgroup of (M, +) which is closed under left R-multiplication. By an R-submodule of M we shall mean a normal subgroup A of (M, +) satisfying r(m + a) — rm ( A for all r ( R, m ( M, a ( A. Submodules of RR are left ideals of R. If A is a left ideal of R and AR = {ar | a ( A,r ( R} ⊂ A, then A is an ideal of R. For any subset B of an R-module M the set {r ( R | rB = (0)} is called the left annihilator of B, denoted by l(B). If B = {x} we write l(x) instead of l({x}). For all subsets B of RM, l(B) is a left ideal of R. If B is an R-subgroup, l(B) is an ideal. Near-ring homomorphisms and R-homomorphisms (that is, near-ring module homomorphisms) are defined in the usual manner. The set of all R-homomorphisms between left R-modules M and N will be denoted by HomR(M, N). An R-homomorphism f: M → N is normal if f(M) is an .N-submodule of N. An exact sequence splits if there exists a normal M such that. The short exact sequence (s.e.s.) splits if the sequence splits. The exact sequence splits if there exists such that The exact sequence splits if and only if the s.e.s. splits [2, Lemma 2.1]. If L and N are submodules of M such that M = L + N and we write A left R-module M is monogenic if there exists a | M such that Ra = M; M is right cancellative if for each nonzero m | M and the identity implies. Thus R is right cancellative if RR is right cancellative. An R-module M is called irreducible if it has no proper nonzero R-subgroups. M is simple if it contains no proper nonzero R-submodules. M is semisimple (also called completely reducible) if M is a direct sum of simple submodules. A near-ring R is called semisimple if RR is semisimple. Therefore, a near-ring R is semisimple if and only if R is the direct sum of simple left ideals. A module M is said to be the semi-direct sum of its R-subgroups A and B, and write M = A∔B if A is an R-submodule, M = A + B and A ⋂ B = (0). In this case B is called a semi-direct summand of M. Then every m ∈ M can be expressed uniquely as a + b for some a∈ A, b ∈ B. Moreover the canonical projection p : M → B given by p(a + b) = b is an R-homomorphism (see Mason [2, p. 46]), R is called strictly semisimple if R is a direct sum of irreducible left ideals. As noted in [2, Theorem 3.5], R is strictly semisimple if and only if every R-subgroup of a monogenic R-module is a semi-direct summand (equivalently, if L1,L2 are R-subgroups of R and 0 → L1 → L2 is exact, then it splits). A near-ring R is regular if for each a ∈ R, ∃ x ∈ R such that a = axa. It can be shown that R is regular if and only if, for all a ∈ R, ∃ an idempotent e = e2 ∈ R such that Ra = Re [3, p. 346]. The socle of R is the sum of all minimal left ideals of R and is denoted by soc(R). For undefined terms and notations used in the sequel, we refer to Pilz [3]