Talk:Multiplicatively closed set

Zero should not be excluded

Most references by notable authors (Atiyah and Macdonald, Bourbaki, Eisenbud, Lang, Matsumura) do not forbid S to contain 0, so I think that we should not either. Localization makes sense even if S contains 0; it is just that the result is the zero ring. Allowing such localizations simplifies theorem statements. For instance, there is a theorem that if A is a commutative ring, and f and g are elements of A, then $A[1/f]\otimes _{A}A[1/g]\simeq A[1/(fg)]$. It would be sad to have to restrict this theorem by saying that we cannot use it, for example, when f and g are nonzero elements with fg=0. Ebony Jackson (talk) 02:04, 16 November 2013 (UTC)