Abstract

The notion of Krull dimension of a ring (commutative with identity) is of great importance in the field of commutative algebra. In particular, the study of dimension behavior in ring extensions is a central issue. More precisely, we may ask if J? C T is an extension, what is the relationship, if any, between dim(i?) and dim(T)? Two fundamental types of ring extensions are integral extensions and polynomial extensions. It is known that if _R C T is an integral extension, then the extension is going-up (GU) and has the "incomparable" property (INC), which implies that dim(i?) = dim(r). The polynomial case is more interesting. It is a classical theorem in commutative algebra that if dim(i?) = n then we have the bounds

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