Goldman's primary decomposition and the tertiary decomposition

Abstract

Emmy Noether’s theory of the primary decomposition of a submodule of a finitely generated module over a commutative noetherian ring was generalized for modules over a not necessarily commutative left noetherian ring R by Lesieur and Croisot [4], and recently by 0. Goldman [2]. In this note the relations between Goldman’s primary decomposition theory and the tertiary decomposition theory of Lesieur and Croisot are studied. It is shown that each finitely generated Goldman-primary R-module is tertiary (Corollary 2.2). The converse does not hold by Remark 2.3. In fact, each finitely generated tertiary R-module is Goldman-primary if and only if nonisomorphic, indecomposable, injective R-modules have different associated prime ideals (Theorem 2.4). On the other side by Proposition 1.1 there always exists a natural one-to-one correspondence between the set r of all isomorphism classes of indecomposable, injective R-modules and the set p of all prime kernel functors of the category $01 of all R-modules. As an application of the above results a short proof of the following theorem is given which, because of Theorem 2.4, is equivalent to a theorem of Gabriel [2]: If each finitely generated tertiary R-module over the left noetherian ring R is Goldman-primary, then the Krull-dimension of R in the sense of Gabriel-Rentschler [A coincides with the usual Krull-dimension of R, if it is finite (Theorem 3.4). All rings R considered in this note have an identity element. Each R-module is a unitary left R-module. E(M) always denotes the injective envelope of the R-module M. Concerning the terminology we refer to Lesieur and Croisot [4] and Herstein [3].