If $R$ is a local ring with maximal ideal $m$ and $I$ is an ideal, then if $R/I$ is a vector space over $R/m$? If it's not true, under what condition for the ideal $I$, do we have $R/I$ is a vector space over $R/m$?
2026-05-06 08:46:40.1778057200
Quotient ring of a local ring
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No. It is true that $R/m$ is an $R/I$-module, but $R/I$ is not an $R/m$-module (=vector space because $R/m$ is a field).