zerodivisor. Thus, any proper ideal I of Q(A) is contained in the set of zerodivisors of Q(A); that set equals the union of the minimal prime ideals since Q(A) is reduced. By prime avoidance, I must be contained in some . Hence, the ideals are maximal ideals of Q(A). Also, their intersection is zero. Thus, by the Chinese remainder theorem applied to Q(A), . Let S be the multiplicatively closed set of non-zerodivisors of A. By exactness of localization, , which is already a field and so must be . Generalization If is a commutative ring and is any multiplicatively closed set in , the localization can still be constructed, but the ring homomorphism from to might fail to be injective. For example, if , then is the trivial ring. Citations References Commutative algebra Ring theory de:Lokalisierung_(Algebra)#Totalquotientenring