Narrow Results

Let $R$ be a commutative ring with identity, $X$ be an indeterminate over $R$, $R[X]$ be the polynomial ring over $R$, $R[[X]]$ be the power series ring over $R$, and $Q$ be a principal prime ideal of $R[X]$ with $(Q\cap R)[X]⊊Q$. It is well known that if $R$ is an integral domain, then $R{[X]}_Q$ is a DVR and $R[X]$ has infinitely many such principal prime ideals. In this paper, among other things, we show that (i) $R_{Q\cap R}$ is a field, (ii) $R{[X]}_Q$ is a DVR, but (iii) there is a ring $R$ such that $R[X]$ has no principal prime ideal. We also study the maximal ideals of $R[[X]]$ that are principal.

A domain R is a maximal non-ACCP subring of its quotient field if and only if R is either a two-dimensional valuation domain with a DVR overring or a one-dimensional nondiscrete valuation domain. If R ⊂ S is a minimal ring extension and S is a domain, then (R,S) is a residually algebraic pair. If S is a domain but not a field, a maximal non-ACCP subring extension R ⊂ S is a minimal ring extension if (R,S) is a residually algebraic pair and R is quasilocal. Results with a similar flavor are given for domains R ⊂ S sharing a nonzero ideal, with applications to rings R of the form A + XB[X] or A + XB[[X]]. If R ⊂ S is a minimal ring extension such that R is a domain and S is not (R-algebra isomorphic to) an overring of R, then R satisfies ACCP if and only if S satisfies ACCP.