A ring related to a given associative ring with an identity. The (right classical) ring of fractions of is the ring in which every regular element (that is, not a zero divisor) of is invertible, and every element of has the form with . The ring exists if and only if satisfies the right-hand Ore condition (cf. Associative rings and algebras). The maximal (or complete) right ring of fractions of is the ring , where is the injective hull of as a right -module, and is the endomorphism ring of the right -module . The ring can also be defined as the direct limit where is the set of all dense right ideals of (a right ideal of a ring is called a dense ideal if

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This notion is also called a ring of quotients.