Is it true that the cardinality of every maximal linearly independent subset of a finitely generated free module $A^{n}$ is equal to $n$? n$ (not just at most $n$, but in fact $n$)? Here $A\neq 0$ A$ is a nonzero commutative ring. I know that it's true if $A$ is Noetherian or integral domain. I thought it was not true in general but I came up with something that looks like a proof and I can't figure out where it went wrong.
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hopefully made title less misleading; edited title
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Rank Linearly independent subsets of a free module |
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Is it true that the cardinality of every maximal linearly independent subset of a finitely generated free module $A^{n}$ is $n$? Here $A\neq 0$ is a commutative ring. I know that it's true if $A$ is Noetherian or integral domain. I thought it was not true in general but I came up with something that looks like a proof and I can't figure out where it went wrong. |
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