NOTES ON DIFFERENTIAL IDEALS OF LASKERIAN RINGS

Abstract

Let $R$ be a ring and $d$ a derivation of $R$. We consider the following three conditions: (a) every quasi-prime $d$-ideal of $R$ is prime, (b) any weak associated prime of every $d$-ideal of $R$ is a $d$-ideal and (c) every $d$-prime $d$-ideal of $R$ is prime. In this paper we show that if $R$ is a Laskerian ring, then the two conditions (a) and (b) are equivalent. Furthermore we show that if $R$ is a strongly Laskerian ring, then any $d$-prime $d$-ideal of $R$ is quasi-prime, and then the three conditions (a), (b) and (c) are equivalent.