On divided commutative rings

Abstract

Let R be a commutative ring with identity having total quotient ring T. A prime ideal P of R is called divided if P is comparable to every principal ideal of R. If every prime ideal of R is divided, then R is called a divided ring. If P is a non principal divided prime, then P-' = { x 6 T : xP c P ) is a ring. We show that if R is an atomic domain and divided, then the Krull dimension of R 5 1. Also, we show that if a finitely generated prime ideal containing a non zero divisor of a ring R is divided, then it is maximal and R is quasilocal.