A study of S-primary decompositions
摘要
Let R be a commutative ring with identity, and let S ⊆ 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 a finite intersection of S-primary ideals. First we provide an example of an S-Noetherian ring in which an ideal does not have a primary decomposition. Then our main aim 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.