Injective Dimension in Noetherian Rings

Introduction. Among Noetherian rings quasi-Frobenius rings are those which are self injective 10, Theorem 18. This paper is concerned primarily with Noetherian rings whose self injective dimension is finite. Thus, for example, Theorem 3. 3, describing rings of self injective dimension one, can be regarded as a one dimensional analogue of the theory of quasi-Frobenius rings. Integral group rings furnish a basic example here. This theorem, noticed independently by Jans 12, was suggested to the author by a problem on torsion free modules to which we apply it in 5, and it is the origin of the present paper. The balance of the paper consists essentially of elaborations on various aspects of the proof of Theorem 6. 3, which characterizes commutative Noetherian rings of finite self injective dimension (2). They are a special class of Macaulay-Cohen rings (see 5 for definition) and enjoy most of the visible properties, locally, of local complete intersections, the latter constituting a fundamental example. For local rings we have, thus, the following hierarchy: regular => local complete intersection => finite self injection dimension => Macaulay-Cohen;and none of the implications can be reversed. 1. The chain condition. Injective modules are seldom finitely generated so, when considering them, the conventional uses of chain conditions on the ring are not available. Theorem 1. 1 provides formulations of the ascending chain condition which are better adapted for our purposes. The theorem is stronger than necessary for our applications, but it may be of interest for suggesting possibly useful definitions of the chain condition in more general abelian categories. We begin by recalling the basic facts about injective envelopes. A is a ring and all modules are left A-modules. A monomorphism 0-*A-*E of A-modules is called essential (we also say that A is an essential submodule of E) if A (~ = Q=*B = 0 for all submodules B of E. If, in addition, E is injective, we call 0->. 4->E an injective envelope of A, and we sometimes indicate this by writing E (A) for E (the embedding being understood, but undenoted).