Research in Algebra

Module Theory

There are five main topics of current interest, as follows (in no particular order):

1. Supplemented modules

Given a submodule N of a module M over a general ring R there exists a submodule K of M which is maximal with respect to the property that it intersects K  in  0. Such a submodule K is called closed (in M). However, in general there need not exist a submodule L minimal with respect to the property M = N+L. If such an L does exist then it is called a supplement of N (in M) and if every submodule of M has a supplement then M is called supplemented. For example, Artinian modules are supplemented. In a series of papers from 1974, ZÖschinger considered the class of supplemented modules both in general and in special cases. Recently there has been an upsurge of interest in this topic and new information has been obtained.

2. Prime submodules

Let R be a commutative ring. Let M be a finitely generated R-submodule. A submodule K of M is prime if K is not equal to M and, for in  R and m in M, rm is in K only if  m is in K or rM is contained in K. In case M = R, the prime submodules of M are precisely the prime ideals. Prime ideals in commutative rings have been extensively studied over many years and from several different standpoints. Prime submodules have been studied more recently and it has been discovered that the collection of prime submodules of M can be very complicated and can exhibit unexpectedly bad behaviour. There are many fundamental questions with only partial answers.

3. c-injectivity

Let R be any ring and let M be a right R-module. The module M is called extending (or a CS-module) if every closed submodule is a direct summand. For example, injective, or more generally quasi-injective, modules are extending. There is a large collection of data about extending modules, although not every question has been resolved. Much less well known are c-injective modules. There are modules M with the property that every homomorphism from a closed submodule K of M to M can be lifted to M. Extending modules are c-injective, but not conversely. It has been shown that c-injective modules share some of the properties of extending modules but much work remains to be done to understand this class of modules.

4. The Z* functor

The functor Z* has been around for 30 years but has only recently received much attention. Given a general module M, Z*(M) is the intersection of M and all maximal submodules of the injective hull of M. (There is a more natural definition of Z*(M)!) If R is a left Noetherian ring satisfying a polynomial identity then Z*(M) = {m in M: m S = 0} for any right R-module M, where S is the left socle of R. The question arises whether Z* can be described so sucessfully for a wider class of rings.

5. Modules satisfying n-acc

Given a positive integer n, a module M satisfies n-acc if every ascending chain of n-generated submodules terminates. If R is a semiprime right Noetherian ring then every free right R-module satisfies n-acc, for every positive integer n. However, there exists a right Noetherian ring R such that every non-finitely generated free right R-module does not satisfy 1-acc. A famous question of P.M. Cohn remain unanswered, namely if R is a ring and n a positive integer such that the free right R-module R satisfies n-acc does every free right R-module satisfy n-acc. Partial results have been obtained.