Localization (commutative algebra): Difference between revisions

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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''.

For example the ring [[modular arithmetic|'''Z'''/''n''<b>'''Z</b>''']] where ''n'' is [[composite number|composite]] is not an integral ___domain. When ''n'' is a [[prime number|prime]] power it is a finite [[local ring]], and its elements are either units or [[nilpotent]]. This implies it can be localized only to a zero ring. But when ''n'' can be factorised as ''ab'' with ''a'' and ''b'' [[coprime]] and greater than 1, then '''Z'''/''n''<b>'''Z</b>''' is by the [[Chinese remainder theorem]] isomorphic to '''Z'''/''a''<b>'''Z</b>''' &times; '''Z'''/''b''<b>'''Z</b>'''. If we take ''S'' to consist only of (1,0) and 1 = (1,1), then the corresponding localization is '''Z'''/''a''<b>'''Z</b>'''.

Two classes of localizations occur commonly in commutative algebra and algebraic geometry: