Valuation and pseudovaluation subrings of an integral domain

摘要

Let R, S be two rings. We say that R is a valuation subring of S (R is a VD in S , for short) if R is a proper subring of S and whenever x is an element of S , we have x is an element of R or x(-1) is an element of R . We denote by Nu (R) the set of all nonunit elements of a ring R . We say that R is a pseudovaluation subring of S (R is a PV in S , for short) if R is a proper subring of S and x(-1) a is an element of R , for each x is an element of S backslash R, a is an element of Nu (R). This article deals with the study of valuation subrings and pseudovaluation subrings of a ring; interactions between the two notions are also given. Let R be a PV in S; the Krull dimension of the polynomial ring on n indetrminates over R is also computed.