ABSTRACT
Considering examples, by the GilmerLParker Theorem for D = K a field and S the factorial (or free) semigroup N" (N = {0,1,2,...}), the semigroup ring K[S\ —
K[Xi,...,Xn] (i.e., the ring of all polynomials in n indeterminates over K) is factorial. As soon, however, as the semigroup S is not isomorphic to some N", the semigroup ring K[S] is not factorial. A first interesting case, in one dimension, is a numerical semigroup S (i.e., S ^ N and S — S = Z). Whatever the integral domain D, the semigroup ring D[S] is never factorial and, in particular, for D = R and S = N \ {1} it shows a very bad factorization behavior. In higher dimensions, for Diophantine monoids S = {x £ N™ Ax = 0} where A € Zmxn, a variety of factorization behaviors can be observed (see Section 3). In general, the semigroup ring K[S] for a field K is not factorial, but factorization behavior can be nice in that it is inside factorial as, for example when S — {x € N3 Ixi + 5z2 — ^ x3}.
