REMARKS ON PRIME IDEALS AND REDUCED RINGS

Abstract

We first prove that a ring $R$ is prime if and only if $a=0$ whenever $a I a=0$ for a nonzero proper ideal $I$ of $R$ and $a \in R$. Then, by using this result, we give another proof of the well known fact that $R$ is prime if and only if the polynomial ring over $R$ is prime, We also prove that if $P$ is an ideal of $R$ maximal with respect to the property that $P$ is disjoint from $S$, then $P$ is a prime ideal of $R$, where $R$ is a ring and $S \subseteq R$ is an $m(I)$-system for a nonzero proper ideal $I$ of $R$. A ring $R$ is called trivial if $R$ has no proper subrings with the identity of $R$. It is proved that a trivial ring $R$ is isomorphic to either $\mathbb{Z}$ (when $\operatorname{ch}(R)=0$ ), $\mathbb{Z}_m$ (when $\operatorname{ch}(R)=m \neq 0$ and the primary decomposition of $m$ is square free), or $\mathbb{Z}_n$ (when $\operatorname{ch}(R)=n \neq 0$ and the primary decomposition of $n$ is not square free), where $\operatorname{ch}(R)$ is the characteristic of $R$.

Received: December 9, 2024
Accepted: March 12, 2025