The lattice of submodules of a module over a noncommutative ring

Abstract

Introduction and summary. Of fundamental importance to the theory of ideals in commutative rings satisfying the ascending chain condition for ideals has been the works of Noether [6] and Krull [4]. What follows, in the first two sections, is an attempt to extend more of the Krull-Noether theory of commutative rings to one-sided ideals in noncommutative rings. A decomposition theory for one-sided ideals in the noncommutative case was provided by Fitting [1]. Although Fitting was able to prove uniqueness theorems in certain cases, he did not prove uniqueness theorems for arbitrary noncommutative rings satisfying the ascending chain condition for right ideals. By methods based on the works of Noether and Fitting, we shall prove decomposition and uniqueness theorems for arbitrary noncommutative rings with identity that satisfy tae ascending chain condition for right ideals. Actually, the theorems in this paper are proved for what we define to be A-R modules, which is a generalization of noncommutative rings. In the third section, we discuss completely indecomposable A-R modules, which follow along the lines of a paper by Snapper [7 ]. A sufficient condition for two faithful representations of a noncommutative ring with identity to be equivalent follows from this discussion. The author is indebted to Professor C. W. Curtis for valuable suggestions in the preparation of this paper. 1. A property of the A.C.C. for A-R modules. We begin by defining an A-R module DEFINITION 1.1. An A-R module is a system consisting of an additive abelian group M, a ring A containing the identity element 1, a ring R containing the identity element e, and two functions defined on the product sets (MXA) and (MXR) having values in M, such that if xa and ya denote the elements in M determined by the elements x, y in M, a in A, and a in R, then