Decomposition Theories for Modules

Introduction. Lesieur and Croisot in [7] have generalized the classical primary decomposition theory for Noetherian modules over commutative rings to the tertiary decomposition theory for Noetherian modules over rings, which are not necessarily commutative, but which have a certain chain condition on ideals. Riley has shown in [8] that for finitely generated unitary modules over left Noetherian rings with identities, the tertiary decomposition theory—in a certain sense—is the only natural generalization of the classical primary theory. The purpose of this paper is to show that the tertiary decomposition theory extends to a larger family of modules. We call this family the family of /-worthy modules. In particular we show that for an arbitrary ring A, the tertiary theory holds for any A-module M which has the property that each factor module of M is finite dimensional in the sense of Goldie. For the family of /-worthy modules we show that, with certain reasonable assumptions, the tertiary theory is the only theory that provides all the salient features of an ideal theory in the sense of the classical primary theory. In order to make this extension, we introduce the abstract concepts of radical functions, associated ideal functions, and decomposition theories. From a radical function we construct a decomposition theory on a certain family of modules. Moreover we show that any decomposition theory can be constructed in this way from a radical function. Therefore our technique appears to be the natural way to construct decomposition theories. Hence we proceed by constructing decomposition theories in this way from various radical functions. In §5 we construct a decomposition theory from the tertiary radical function, t, and prove that it is the tertiary decomposition theory. Moreover we show that a necessary and sufficient condition for the tertiary decomposition theory to exist on an admissible family C of modules is that each module in C be /-worthy. Goldie's dimension theory is then used to produce examples of admissible families of /-worthy modules. An example is given in §6 to show that the associated ideals, which are used to produce the extension of the tertiary theory, need not be prime ideals. Also it is shown that in certain special cases, our definition of associated ideal specializes to the usual definitions of associated prime ideal.