Submodule Lattices of Direct Sums

This paper deals with the poset of submodules of the direct sum of two modules. A well-known theorem of Remak (1930) gives a description of the set of submodules of the direct sum in terms of isomorphisms between sections of the direct summands. Given two submodules described in these terms, it is difficult in general to determine whether one is contained in the other. In this paper, we show how to determine whether one is contained in the other as maximal submodule. That is, we give a method of describing the edge set of the Hasse diagram of this poset.