Let $R$ be a commutative ring without identity. Suppose $R$ doesn't contain a proper maximal ideal, and $R$ is not the zero ring. Prove that $\forall x \in R$, the ideal $xR$ is proper.
2026-03-30 00:05:36.1774829136
Prove that for all $x \in R$, the ideal $xR$ is proper.
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HINT: Suppose that $xR=R$. There is some $y\in R$ such that $xy=x$. Now let $r\in R$ be arbitrary. Then $r=xz$ for some $z\in R$, so $r=xyz=\ldots\;$?