Commutative rings I

Let R be a commutative ring with identity and 
$$S\subseteq R$$

 a multiplicative subset. In this paper we introduce the concept of S-prime ideal which is a generalization of prime ideal. Let P be an ideal of R disjoint with S. We say that P is an S-prime ideal of R if there exists an 
$$s\in S$$

 such that for all 
$$a,b\in R$$

 if 
$$ab\in P,$$

 then 
$$sa\in P$$

 or 
$$sb\in P$$

. We show that S-prime ideals enjoy analogs of many properties of prime ideals and we study them over S-Noetherian rings.

S -prime ideals of a commutative ring

Let D be an integral domain, t be the so-called t-operation on D; and S be a (not necessarily saturated) multiplicative subset of D. In this paper, we study the Nagata ring of S-Noetherian domains and locally S-Noetherian domains. We also inves- tigate the t-Nagata ring of t-locally S-Noetherian domains. In fact, we show that if S is an anti-archimedean subset of D, then D is an S-Noetherian domain (respectively, locally S-Noetherian domain) if and only if the Nagata ring D(X)N is an S-Noetherian domain (respectively, locally S-Noetherian domain). We also prove that if S is an anti-archimedean subset of D, then D is a t-locally S-Noetherian domain if and only if the polynomial ring D(X) is a t-locally S-Noetherian domain, if and only if the t-Nagata ring D(X)Nv is a t-locally S-Noetherian domain.

https://www.koreascience.or.kr:443/article/JAKO201530861235149.pdf

A Note on S-Noetherian Domains

On s -strong mori domains

In this paper we study right S-Noetherian rings and modules, extending notions introduced by Anderson and Dumitrescu in commutative algebra to noncommutative rings. Two characterizations of right S...

On right S-Noetherian rings and S-Noetherian modules

Let R be a commutative ring with identity and S a multiplicative subset of R. We say that R is an S-Noetherian ring if for each ideal I of R, there exist an s ∈ S and a finitely generated ideal J of R such that sI ⊆ J ⊆ I. In this article, we study transfers of S-Noetherian property to the composite semigroup ring and the composite generalized power series ring.

S-Noetherian Properties of Composite Ring Extensions

Let R be an integral domain, Γ be a nonzero torsion-free commutative cancellative monoid, t be a twist function of Γ on R, R[X;Γ] be the semigroup ring of Γ over R, and Rt[X;Γ] be the twisted semig...

Dong Yeol Oh

Papers

S-Noetherian Properties of Composite Ring Extensions

Divisibility properties of twisted semigroup rings