Valuation and pseudovaluation subrings of an integral domain

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 S , we have x R or x -1 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 R , for each x S R, a 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.