Notes on Noetherian Semigroup Rings

Let A be a commutative ring with identity and let S be a torsion-free canoellative commutative semigroup with identity. We write the semigroup operation on S as addition and assume that S_??_{0}. We consider the semigroup ring A[X; S] of S over A. In [12] we determined conditions under which A[X; S] is a Noetherian ring. We concern with A[X; S] further as a Noetherian ring in this paper. Throughout this paper A denotes a commutative ring with identity. S denotes the above mentioned semigroup. The smallest group containing S is called quotient group of S and is denoted by q(S). G denotes the quotient group of S.