FINITE DIRECT SUMS CONTROLLED BY FINITELY MANY PERMUTATIONS

The classes of uniserial modules, biuniform modules, cyclically presented modules over a local ring, more generally, couniformly presented modules, and kernels of morphisms between indecomposable injective modules, are some among the classes of modules which are characterized by a pair of invariants. These invariants also completely describe when finite direct sums of such modules are isomorphic. In this paper, we are interested in modules characterized by finitely many invariants and in their finite direct sums. We give a general criterion to produce classes S of such modules, and we completely describe how modules satisfying said criterion can be grouped together to form isomorphic finite direct sums. The connection between the regularity of finite direct sums of modules in S and a certain associated hypergraph H(S) is also investigated.