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Let $R$ be a commutative ring with $1$, and let $S$ be the set of elements that are not zero divisors in $R$; then $S$ is a multiplicatively closed set. Hence we may localize the ring $R$ at the set $S$ to obtain $S^{-1}R$. This new ring is called the Total ring of fractions.

we may also localize the R-module M to obtain $S^{-1}M$. Is there a special name for this concept?

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For an $R$-module $M$, the localization $S^{-1}M$ is usually called the module of fractions with respect to $S$. If $S$ is the set of all non-zerodivisors in $R$, one often calls $S^{-1}M$ the total quotient module. This is simply the module localization compatible with the total quotient ring

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  • $\begingroup$ tnx. can u please give link that uses the terminology "total quotient module". $\endgroup$ Commented Nov 7 at 12:10
  • $\begingroup$ It is worth noting that this module is often zero unless $M$ is a faithful module. In particular, if $\operatorname{Ann}_R(M)$ contains a non-zerodivisor, then $S^{-1}M$ vanishes. $\endgroup$ Commented Nov 7 at 17:35

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