$R$ is a commutative ring. $I$ and $J$ are ideals of $R$ with $J\subseteq I$. Assume that in the quotient ring $R/J$, $I/J$ is a prime ideal. Is the ideal $I$ a prime ideal in $R$?
2026-04-04 05:43:19.1775281399
Assume that $I/J$ is a prime ideal of $R/J$, is $I$ a prime ideal of $R$?
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Yes, just compute the quotient, if $J\subseteq I$ then $I/J\subseteq R/J$ is prime means $(R/J)\bigg/ (I/J)$ is an integral domain, but the second isomorphism theorem says
so $I$ must be a prime ideal of $R$.