Note on the unit group of R[X;S], II

Let R be a commutative ring, and let S be a commutative semigroup. We study a semigroup version of Karpilovsky's Problem (K, chapter 7, problem 9) concerning the unit group of a group ring. We give a preciser decomposition theorem for the unit group of a semigroup ring. This is a continuation of our (M3). Thus a submonoid S of a torsion-free abelian (additive) group is called a grading monoid (or a g-monoid). Throughout the paper we assume that S is non-zero. We consider the semigroup ring R(X;S) of S over a commutative ring R. We denote the unit group of S by H=H(S). We denote the nilradical of R by N=N(R), and let U=U(R) be the unit group of R. The group of units f=Σ ααXα of R(X;S) with Σ αα=1 is denoted by