On Graded Rings, II ($\mathbf{Z}^n$-graded rings)

Abstract

The main purpose of our paper is to give a purely ring-theoretic proof of the Hochster's result on normal semigroups and, applying our way of proof further to arbitrary affine semigroups $S$ , to find a criterion of the ring $k[S]$ to be Macaulay (resp. Gorenstein) in terms of $S$ . Note that this was achieved by the authors and Suzuki