A Study of S-Primary Decompositions
Abstract
Let R be a commutative ring with identity and S ⊂eq R be a multiplicative set. An ideal Q of R (disjoint from S) is said to be S-primary if there exists an s∈ S such that for all x,y∈ R with xy∈ Q, we have sx∈ Q or sy∈ rad(Q). Also, we say that an ideal of R is S-primary decomposable or has an S-primary decomposition if it can be written as finite intersection of S-primary ideals. In this paper, first we provide an example of S-Noetherian ring in which an ideal does not have a primary decomposition. Then our main aim of this paper is to establish the existence and uniqueness of S-primary decomposition in S-Noetherian rings as an extension of a historical theorem of Lasker-Noether.
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