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If an $R$-module $M$ is generated by $n$ elements then how to show that $M$ can be realised as a quotient of $R^n$?

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Hint: Let $x_1,\ldots,x_n$ be generators of $M$, and define $f:R^n\to M$ by $$f(r_1,\ldots,r_n)=r_1x_1+\cdots+r_nx_n.$$ This is a surjective $R$-module homomorphism from $R^n$ to $M$. Is there a homomorphism theorem you can apply here to realize $M$ as a quotient of $R^n$?

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