Question about exact sequence of modules

abstract-algebra commutative-algebra exact-sequence abelian-categories

Let $R\to S$ be a ring homomorphism, i e. $S$ is a $R$-algebra. If $L,M,N$ are $S$-modules, they are automatically $R$-modules too. If I know that $0\to L\to M\to N\to 0$ is an exact sequence of $S$-modules, can I deduce that it is exact as a sequence of $R$-modules? Or the viceversa?


Solution 1:

Note that exactness is defined in purely set theoretic terms: the image of one map is the preimage of 0 along another map. So if you are given a sequence of $S$-modules, it is exact if and only if it is exact as sequence of $R$-modules. But for this to hold, you necessarily have to know that you started with a sequence of $S$-modules, since an $R$-linear map of $S$-modules need not be $S$-linear!

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