Rings Whose Nonzero Modules Have Maximal Submodules

Abstract

All rings are assumed to be associative and (except for nil-rings and some stipulated cases) to have nonzero identity elements. Expressions such as a “Noetherian ring” mean that the corresponding right and left conditions hold. A module is said to be simple if it does not have nonzero proper submodules. A submodule N of a module M is called a maximal submodule (in M) if the factor module M/N is simple. A ring A is called a right max ring (or a right Bass ring) if every nonzero right A-module has a maximal submodule. A ring is said to be orthogonally finite if it does not contain an infinite set of nonzero orthogonal idempotents. Bass [11] proved that a ring with the minimum condition on principal left ideals is an orthogonally finite right max ring. Cozzens [25] and Koifman [61] constructed examples of orthogonally finite right max rings which do not satisfy the minimum condition on principal left ideals. This review contains some new and some old results related to max rings. We present the necessary notation and definitions. Let A be a ring. We denote the center of the ring A by C(A). A subset B of A is said to be central (in A) if B ⊆ C(A). If M and N are two modules, then Hom(M,N) denotes the Abelian group formed by all homomorphisms M → N . For a module M , we denote by End(M) the endomorphism ring Hom(M,M) of M . For a subset B of A, the right annihilator and the left annihilator of B in A are denoted by rA(B) and A(B), respectively. We can omit the subscripts if the situation is obvious. An element a of A is said to be right regular (resp. left regular) in A if r(a) = 0 (resp. (a) = 0). A ring A is said to be regular if for every element a of A, there exists an element b of A such that a = aba. A ring A is said to be right weakly regular if B2 = B for every right ideal B of A. A ring without nonzero nilpotent ideals is called a semiprime ring. A ring is called a prime ring if the product of any two of its nonzero ideals is not equal to zero. A ring A is said to be simple if each of its nonzero ideals coincides with A. A ring is called a domain if the product of any two of its nonzero elements is not equal to zero. For a ring A, an ideal P of A is called a prime (resp. semiprime and completely prime) ideal if the factor ring A/P is a prime ring (resp. a semiprime ring and a domain). A ring A is called a P.I.-ring (or a ring with polynomial identity) if A satisfies the polynomial identity f(x1, . . . , xn) = 0, where f(x1, . . . , xn) is a polynomial in noncommutative variables with coefficients in the ring of integers Z, and Z coincides with its ideal generated by the coefficients of f(x1, . . . , xn). A submodule N of a module M is said to be essential (in M) if N has nonzero intersection with any nonzero submodule of the module M . In this case, we say that M is an essential extension of the module N . A submodule N of a module M is called a superfluous submodule (in M) if N +M ′ = M for every proper submodule M ′ of M .