Let C ⊂ N be an affine semigroup, and R = K[C] its semigroup ring. This paper is a collection of various results on “C-graded” R-modules M = ⊕ c∈C Mc, especially, monomial ideals of R. For example, we show the following: If R is normal and I ⊂ R is a radicalmonomial ideal (i.e., R/I is a generalization of Stanley-Reisner rings), then the sequentially Cohen-Macaulay property of R/I is a topologi...