localization

Let $R$ be a commutative ring and let $S$ be a nonempty multiplicative subset of $R$ . The localization of $R$ at $S$ is the ring $S^{-1}R$ whose elements are equivalence classes of $R\\times S$ under the equivalence relation $(a,s)\\sim(b,t)$ if $r(at-bs)=0$ for some $r\\in S$ . Addition and multiplication in $S^{-1}R$ are defined by:

•
$(a,s)+(b,t)=(at+bs,st)$
•
$(a,s)\\cdot(b,t)=(a\\cdot b,s\\cdot t)$

The equivalence class of $(a,s)$ in $S^{-1}R$ is usually denoted $a/s$ . For $a\\in R$ , the localization of $R$ at the minimal multiplicative set containing $a$ is written as $R_{a}$ . When $S$ is the complement of a prime ideal $\\mathfrak{p}$ in $R$ , the localization of $R$ at $S$ is written $R_{\\mathfrak{p}}$ .

单词	Localization
释义	localization Let $R$ be a commutative ring and let $S$ be a nonempty multiplicative subset of $R$ . The localization of $R$ at $S$ is the ring $S^{-1}R$ whose elements are equivalence classes of $R\\times S$ under the equivalence relation $(a,s)\\sim(b,t)$ if $r(at-bs)=0$ for some $r\\in S$ . Addition and multiplication in $S^{-1}R$ are defined by: • $(a,s)+(b,t)=(at+bs,st)$ • $(a,s)\\cdot(b,t)=(a\\cdot b,s\\cdot t)$ The equivalence class of $(a,s)$ in $S^{-1}R$ is usually denoted $a/s$ . For $a\\in R$ , the localization of $R$ at the minimal multiplicative set containing $a$ is written as $R_{a}$ . When $S$ is the complement of a prime ideal $\\mathfrak{p}$ in $R$ , the localization of $R$ at $S$ is written $R_{\\mathfrak{p}}$ .
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