Suppose $R$ be any ring containing left ideal $I$. Then $I$ is submodule of $R$, so $R/I$ is R-module. My question is, is $R/I$ always a finitely generated?
2026-04-04 02:53:16.1775271196
Is quotient module finitely generated?
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Denote $\bar r$ the class of $r\in R$ modulo $I$. Then $$ \bar r=\overline{r\cdot1}=r\cdot\bar1 $$ i.e. $R/I$ is generated as $R$-module by $\bar1$.
More generally, if $M$ is a $R$-module generated by $m_1,...,m_s$, then the classes $\bar m_1,...,\bar m_s$ generate $M/IM$ over $R$.