Answer by Joshua Tilley for If the quotient of an ideal is principal, is the...
No!Let $I=(x)\subset J=(x,y)\subset R=k[x,y]$, then $J/I=(y)$ is a principal ideal of $R/I=k[x,y]/(x)$. Suppose for contradiction that $J=(x,y)$ is principal, equal to $(f)$ for $f\in k[x,y]$. Then...
View ArticleIf the quotient of an ideal is principal, is the original ideal principal?
Let $R$ be a unity conmutative ring and $I \subset J \subset R$ ideals of $R$. Is it true that if $J / I$ is principal, then $J$ is principal? This question has came to me on other excercise in which,...
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