Narrow Results

This paper concerns ring properties which are induced from the structure of the powers of prime ideals. An ideal $I$ of a ring $R$ is called $n$-primary (respectively, $T$-primary) provided that $AB\subseteq I$ for ideals $A,B$ of $R$ implies that $(A+I)/I$ or $(B+I)/I$ is nil of index $n$ (respectively, $(A+I)/I$ or $(B+I)/I$ is nil) in $R/I$, where $n\ge 1$. It is proved that for a proper ideal $I$ of a principal ideal domain $R$, $I$ is $T$-primary if and only if $I$ is of the form $p^{k}R$ for some prime element $p$ and $k\ge 1$ if and only if $I$ is $2$-primary, through which we study the structure of matrices over principal ideal domains. We prove that for a $T$-primary ideal $I$ of a ring $R$, $R/I$ is prime when the Wedderburn radical of $R/I$ is zero. In addition we provide a method of constructing strictly descending chain of $n$-primary radicals from any domain, where the $n$-primary radical of a ring $R$ means the intersection of all the $n$-primary ideals of $R$.