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The Ring of Quotients R[S]: R an Integral Domain and S a Commutative Cancellative Semigroup

✍ Scribed by James A. Bate; John K. Luedeman

Publisher
John Wiley and Sons
Year
1983
Tongue
English
Weight
401 KB
Volume
113
Category
Article
ISSN
0025-584X

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✦ Synopsis

The theory of rings of quotients of a given ring has been developed by many authors including Asmo [ l ] , JOHNSON [6], UTUMI [ l l ] , and STENSTEOM [lo]. A corresponding theory of semigroups of quotients has been studied by BERTHUUME [3], MOMORRIS [8], and &KLE [6]. LUEDEMAN [7] has begun the development of a theory of the quotient ring of a semigroup ring R[S]. Using the a-set Z(S) = {J[S]: J E Z and C is a a-set on R } , LUEDEUN has shown that in certain cases &,(,,(R[S]) is a ring isomorphic to Qz(R) [S] where &,,(T) is the UTUMI quotient ring of T over the a-set A. This paper further extends LUEDEMAN'S work by considering the theory of quotient rings of semigroup rings over a a-set Z(A) = { J [ I ] : J € Z, a a-set on R and I € A , a a-set on s}.

I n section 2, conditions under which Z(A) becomes a a-set for R[S] are discussed. In section 3, S is required to be a left Ore monoid with 0 and A consists of left ideals of S containing cancellable elements. Then if R is a ring with identity and Z is a a-set of finite type on R, &L.

(d)(R[S]) is shown to be isomorphic to &(R) [&d(S)].

In section 4, this result is slightly generalized when S is commutative.

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