# 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.

**How to Cite This Entry:**

Fractions, ring of.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Fractions,_ring_of&oldid=17898