SEMIGROUP RINGS THAT ARE INSIDE FACTORIAL AND THEIR CALE REPRESENTATION ULRICH KRAUSE 1. Introduction: Not factorial but still nice? Concerning the ubiquitous property of factoriality, Robert Gilmer and Tom Parker in a beautiful paper proved that a semigroup ring D[S] is factorial pre- cisely if both the integral domain D and the semigroup S are factorial (and S satisfies some additional property which is described in [10, Theorem 7.17] and [11, Theorem 14.16]). Even for a field D, a non–factorial but otherwise nice semigroup S leads, according to the Gilmer–Parker Theorem, to a non–factorial semigroup ring D[S] — which, nevertheless, can be quite nice. In this paper, we address the question of whether the semigroup ring D[S] still enjoys some nice factorization properties — albeit neither D nor S is assumed to be factorial. The answer is given by Theorem 3.2 in section 3 where we show that D[S] is inside factorial if both D and S are inside factorial and D[S] is solid. Before proving this result, we define and examine in Section 2 the concepts of inside factoriality and Cale representation, respectively. But before doing so, we discuss the meaning of “nice” as used above and recall some fundamental notions. Let D denote an integral domain with identity (or just a domain for short). Let S denote an additively written abelian semigroup with 0 which is cancellative ( i.e., x + z = y + z implies x = y where x, y, z ∈ S). Furthermore, assume S to be torsion free (i.e., nx = ny implies x = y for x, y ∈ S and n ≥ 1) and that S contains only trivial invertible elements (i.e., x, −x ∈ S implies that x = 0). For an integral domain D and a semigroup S, the semigroup ring D[S] is the set of all mappings f : S → D with f (s) = 0 only for finitely many s ∈ S equipped with an addition defined by (f + g)(s)= f (s)+ g(s) and a multiplication defined by (f · g)(s)= ∑ t+u=s f (t)g(u). An element f of D[S] we write as ∑ s∈S f (s)X s (Cf. [11]). Considering examples, by the Gilmer–Parker Theorem for D = K a field and S the factorial (or free) semigroup N n (N = {0, 1, 2,... }), the semigroup ring K[S]= K[X 1 ,... ,X n ] (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 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 n | Ax =0} where A ∈ Z m×n , 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 ∈ N 3 | 2x 1 +5x 2 =3x 3 }. 1