Generalized formal power series rings
Let \(R\) be a commutative ring with 1 and \((S,\leq)\) a strictly ordered commutative monoid. Then \(A=[[R^{S,\leq}]]\), the set of functions \(f:S\to R\) such that \(\text{supp}(f)\) is artinian and narrow, is a ring with 1 under pointwise addition and convolution, called the ring of generalized power series. In this paper, the author studies the prime and maximal ideals of \(A\) and completely determines the maximal ideals when \(s\geq 0\) for each \(s\in S\). He determines necessary and sufficient conditions for an \(f\in A\) to be nilpotent when \(\text{char} R\) is nonzero and \(S\) is torsion-free. He also determines some sufficient conditions for \(A\) to be integrally closed when \(R\) is an integral domain and \(S\) is torsion-free and completely integrally closed.
scientific article; zbMATH DE number 1185640