Fractions, ring of

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

References

[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)

Comments

This notion is also called a ring of quotients. For a commutative integral domain we obtain the field of fractions.