Divisibility Properties in Graded Integral Domains
There are many divisibility properties one can consider on an integral domain R. They range from chain conditions and closedness properties to lengths of factorizations and class groups. If, in addition, R is a graded ring, then in some cases we need only consider homogeneous information. Here we investigate to what extent conditions on the monoid S of nonzero homogeneous elements of R or the homogeneous ideals of R determine divisibility properties of R. It is somewhat surprising that in some cases divisibility properties of R are completely determined by S; this depends on the divisibility property considered and how R is graded. Things behave best when R is a Z+-graded integral domain or a semigroup ring.