Fractions, ring of

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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


[1] J. Lambek, "Lectures on rings and modules" , Blaisdell (1966)
[2] V.P. Elizarov, "Quotient rings" Algebra and Logic , 8 : 4 (1969) pp. 219–243 Algebra i Logika , 8 : 4 (1969) pp. 381–424
[3] B. Stenström, "Rings of quotients" , Springer (1975)


This notion is also called a ring of quotients.

How to Cite This Entry:
Fractions, ring of. Encyclopedia of Mathematics. URL:,_ring_of&oldid=17898
This article was adapted from an original article by L.A. Bokut' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article