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For general [[commutative ring]]s, we don't have a field of fractions. Nevertheless, a localization can be constructed consisting of "fractions" with [[denominator]]s coming from ''S''; in contrast with the integral ___domain case, one can now safely 'cancel' from [[numerator]] and [[denominator]] only elements of ''S''.
This construction proceeds as follows: on ''R'' × ''S'' define an [[equivalence relation]] ~ by setting (''r''<sub>1</sub>,''s''<sub>1</sub>) ~ (''r''<sub>2</sub>,''s''<sub>2</sub>) [[iff]] there exists ''t'' in ''S'' such that
:''t''(''r''<sub>1</sub>''s''<sub>2</sub> We think of the [[equivalence class]] of (''r'',''s'') as the "fraction" ''r''/''s'', and using this intuition, the set of equivalence classes ''R*'' can be turned into a ring; the map ''j'' : ''R'' → ''R*'' which maps ''r'' to the equivalence class of (''r'',1) is then a [[ring homomorphism]]. The above mentioned universal property is the following: the ring homomorphism ''j'' : ''R'' → ''R*'' maps every element of ''S'' to a unit in ''R*'', and if ''f'' : ''R'' → ''T'' is some other ring homomorphism which maps every element of ''S'' to a unit in ''T'', then there exists a unique ring homomorphism ''g'' : ''R*'' → ''T'' such that ''f'' = ''g'' o ''j''.
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