Abstract: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 th… Show more
“…(1) ΠieiNi = iV and for no ie / is f\WV, g A^; (2) In the terminology of [1] p is a radical function on ^/έ and P is the associated ideal function on ^/f that is obtained from p. Therefore Theorem 4.10 in [1] shows that a necessary and sufficient condition for M to have the P-decomposition theory is that M be p-worthy.…”
mentioning
confidence: 99%
“…In [1] we introduced a new technique for constructing decomposition theories for modules and we used it to give necessary and sufficient conditions for the Lesieur-Croisot tertiary decomposition theory to exist on a module over an arbitrary ring. By again making use of this technique we have obtained necessary and sufficient conditions for the classical Lasker-Noether primary decomposition theory to exist on a module over an arbitrary commutative ring.…”
mentioning
confidence: 99%
“…In [1], [4], and [7] investigation was conducted into the connection between the existence of the primary decomposition theory and the Artin-Rees property of a module. In § 2 we have obtained the following new results in this connection.…”
mentioning
confidence: 99%
“…See Example 4 in [1]. We proceed with the task of finding necessary and sufficient conditions for the existence of the primary theory.…”
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.