Fractions, ring of

A ring related to a given associative ring $R$ with an identity. The (right classical) ring of fractions of $R$ is the ring $Q_{\mathrm{cl}}(R)$ in which every regular element (that is, not a zero divisor) of $R$ is invertible, and every element of $Q_{\mathrm{cl}}(R)$ has the form $ab^{-1}$ with $a,b \in R$. The ring $Q_{\mathrm{cl}}(R)$ exists if and only if $R$ satisfies the right-hand Ore condition (cf. Associative rings and algebras). The maximal (or complete) right ring of fractions of $R$ is the ring $Q_{\mathrm{max}}(R) = \mathrm{Hom}_H(\widehat R,\widehat R)$, where $\widehat R$ is the injective hull of $R$ as a right $R$-module, and $H = \mathrm{Hom}_R(\widehat R,\widehat R)$ is the endomorphism ring of the right $R$-module $\widehat R$. The ring $Q_{\mathrm{max}}(R)$ can also be defined as the direct limit $$ \lim_{\rightarrow} \mathrm{Hom}(D,R) $$ where $D$ is the set of all dense right ideals of $R$ (a right ideal $D$ of a ring $R$ is called a dense ideal if $$ \forall 0 \neq r_1,r_2 \in R\ \exists r\in R\ (r_1r \neq0,\,r_2r \in D)\ . $$

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This notion is also called a ring of quotients. For a commutative integral domain we obtain the field of fractions.