I know that a module is the ring-theoretic analogue of a vector space. My question is, is every module over the trivial ring $\{0\}$ a singleton set? Or can there be a module over the trivial ring that has higher cardinality than $1$?
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2Have you checked what the axioms would say? – Randall Jun 24 '22 at 20:29
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1You must have $1x=x$ for all $x$, and therefore $x=1x=0x=0$ for all $x$. – Sassatelli Giulio Jun 24 '22 at 20:30
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@SassatelliGiulio that assumes all rings have $1$ (and that all such modules must be unital), which not everyone (including me) assumes. – Randall Jun 24 '22 at 20:30
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I guess it comes down to the OP being more clear: what do you require of rings, and if they must have an identity, must your modules respect that or not? – Randall Jun 24 '22 at 20:32
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1On second thought, I guess it doesn't, because the answer in the linked dupe covers all cases nicely and clearly. – Randall Jun 24 '22 at 20:33
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@Randall Agreed and I guess my assumptions are explicit enough for a comment. – Sassatelli Giulio Jun 24 '22 at 20:37