Rectangular Simplicial Semigroups

In [3] Bruns, Gubeladze, and Trung define the notion of polytopal semigroup ring as follows. Let P be a lattice polytope in R, i. e. a polytope whose vertices have integral coordinates, and K a field. Then one considers the embedding ι : R → R, ι(x) = (x, 1), and chooses SP to be the semigroup generated by the lattice points in ι(P ); the K-algebra K[SP ] is called a polytopal semigroup ring. Such a ring can be characterized as an affine semigroup ring that is generated by its degree 1 elements and coincides with its normalization in degree 1. The object of this paper are the polytopal semigroup rings given by the simplices ∆(λ1, . . . , λn) ⊂ R with vertices