The Ring of Quotients of R[S]; R a Commutative Ring and S a REES Matrix Semigroup over a Semigroup

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

• Publisher: John Wiley and Sons
• Year: 1981
• Tongue: English
• Weight: 483 KB
• Volume: 101
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19811010122

✦ Synopsis

By JAMES A. RATE and JOHN K . LUEDEMAN of Clemson (I7.S.A.) (Eingegangen am 22. 11. 1979) REES matrix semigroups &I= (S, ,I, -1, P) over a semigroup correspond loosely to the n X n matrix rings over it ring R. It is well known that &(R,)x .=(&(R)),,. Moreover, when S is it finite BRANDT semigroup, S = ( G , m , rn, I ) and the semigroup ring R [ S ] x [ R [ G ] ] , . In this case LUEDEMAN [ l o ] showed that I n view of these results, it is natural to study the semigroup of quotients of & ( M ) and to determine the structure of &,,,,(R[M]).

In this paper, the study of the quotient ring problem begun in [2] and [3] will be continued for a semigroup ring R [ M ] where M is a REES matrix semigroup over a semigroup S. For this case, the structure of the quotient objects will be obtained by ohserving the structure of ideals in M and R [ M ] and the appropriate homomorphisms. For a a-set d on M , defined in terms of a a-set on S, the structure of & ( M ) is given explicitly in terms of a product of quotient rings Q 3 ( S ) of S and a function on the index set A . Similarly, the structure of &,,>,(R[M]) is given explicitly in terms of a direct sum of products of quotient rings &,,,,(R[S]) and functions on 11.