Exercise 19, § 1

Let A be a ring, S a subset of A that admits a calculus of left fractions on A (Exercise 18), and $S'$ the smallest subset of A containing 1 and S and stable under multiplication. We denote by $S^{-1}A$ the ring of left fractions of A associated with S and by $\varepsilon : A\rightarrow S^{-1}A$ the canonical homomorphism.

a) Let M be an A-module. We denote the $S^{-1}$ A-module $S^{-1}A\otimes_AM$ by $S^{-1}M$, and we call it the module of left fractions of M associated with S. Prove that every element of $S^{-1}M$ can be written as $\varepsilon (s)^{-1}\otimes x$ with $x\in M$ and $s\in S'$, and that for $x, y$ in M and $s, t$ in $S'$, the relation $\varepsilon (s)^{-1}\otimes x=\varepsilon (t)^{-1}\otimes y$ is equivalent to the existence of a pair $(c, d)$ of elements of A such that we have $cx=dy$ and $cs=dt\in S'$.

b) We denote by $\varepsilon_M$ the canonical A-linear mapping $x\mapsto 1\otimes x$ from M to $S^{-1}M$. Its image generates the $S^{-1}$A-module $S^{-1}M$; its kernel consists of the elements $x\in M$ for which there exists an $s\in S'$ such that $sx= 0$. The mapping $\varepsilon_M$ is injective (resp. bijective) if and only if for every $s\in S$, the homothety $x\mapsto sx$ of M is injective (resp. bijective).

c) We have $S^{-1}M = 0$ if and only if for every $x\in M$, there exists an $s\in S$ such that $sx= 0$.

d) Let N be a submodule of M. The canonical $S^{-1}$A-linear mapping from $S^{-1}N$ to $S^{-1}M$ is injective. $(^*$In other words, the right A-module $S^{-1}A$ is flat.$_*)$ We identify $S^{-1}N$ with its image in $S^{-1}M$ under this mapping. The submodule $\varepsilon^{-1}_M(S^{-1}N)$ of M consists of the elements $x\in M$ for which there exists an $s\in S'$ with $sx\in N$. We say that it is the left saturation of N for S.

e) The mapping $N'\mapsto \varepsilon^{-1}_M(N')$ is an isomorphism from the ordered set of $S^{-1}A$-submodules of $S^{-1}M$ onto the ordered set of A-submodules of M that are left saturated, (i.e., equal to their saturation) for S. The inverse isomorphism is $N\mapsto S^{-1}N$. f ) If the A-module M is Noetherian, then the $S^{-1}$A-module $S^{-1}M$ is Noetherian. If the A-module M is Artinian, then the $S^{-1}$A-module $S^{-1}M$ is Artinian and the mapping $\varepsilon_M: M\rightarrow S^{-1}M$ is surjective.

g) If the ring A is left Noetherian (resp. left Artinian), then so is the ring $S^{-1}A$. h) Formulate the statements analogous to those of this exercise for the modules of right fractions of right modules.