Indecomposable modules: modules with cores

Abstract

We introduce a class of modules, called modules with cores, larger than any previously known fundamental class of indecomposable modules. The core of a module M is the intersection, C(M), of its nonsuperfluous submodules; that is, the intersection of submodules N such that N 4N'= M for some proper submodule N*. Dually the cocore, C°(M), is the sum of the nonessential submodules of M. Any module with a core (meaning C(M) ¥= 0) or a cocore (C°(M) -=hM) is indecomposable. Our main aim is to describe a classification of radical squared zero Artin algebras such that every indecomposable module of finite length has a core or a cocore. (Finite dimensional algebras over a field are the most important examples of Artin algebras.) The classification shows that there are modules having a core and no cocore. This establishes our claim concerning the size of the class of modules with cores even in the restricted setting of radical squared zero Artin algebras of finite representation type—for modules with waists (see [1]) have a core and a cocore and include, as well as serial modules, modules with simple socles or, dually, simple tops. The proof of the following existence theorem is constructive and uses a result announced in [5] concerning indecomposability of amalgamations of basic modules of finite length. (A nonzero module is basic if it is its own core. For modules of finite length this just means that the module has a simple top.)