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i) Let $ M$ be a free $R$-module. By definition $ M = R \oplus R \oplus\cdots\oplus R$ . Can anyone could explain me why $ M = Rx_1 \oplus\cdots\oplus Rx_n$ where $x_1,\ldots,x_n$ elements of $M$. My idea is to prove that $ R \cong Rx_i$. Is thatthe correct solve?

ii)If $k$ is the quotient ring of $R$ ($R$ an intergal domain), $L$ a finite extension of $k$ and $M$ contains elements of $L$. Then the number of copies of $R$ are the same with $[L:k]$ . If yes what is the prove of that?

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For (i), this is just two choices of notation for the same thing. You can write an element of $M$ as $(r_1, r_2, \ldots, r_n)$, or you can write it as $r_1 x_1 + r_2 x_2 + \cdots r_n x_n$. (So, in the previous notation, $x_1 = (1, 0, 0, \ldots, 0)$, etc.)

For (ii), I don't know what you mean by "M contains elements of L."

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