Search

The main purpose of this paper is to generalize a result of T.G. Lucas which states that if R is a reduced commutative ring and M is a flat R-module, then the idealization R ⋉ M is an A -ring if and only if R is an A -ring [14, Proposition 3.5]. In effect, we drop the reduceness hypotheses and prove that, given an arbitrary commutative ring R and any submodule M of a flat R-module F, R ⋉ M is an A -ring (resp., S A -ring) if and only if R is an A -ring (resp., S A -ring). [ABSTRACT FROM AUTHOR]

This article is concerned with the dimension theory of tensor products of algebras over a field k . In fact, we provide formulas for the Krull and valuative dimension of A ⊗k B when A and B are k -algebras such that the polynomial ring A [n] is an AF-domain for some positive integer n . Also, we compute dimv (A ⊗k B ) in the case where A ⊆ B . [ABSTRACT FROM AUTHOR]