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
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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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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.
Fractions, ring of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fractions,_ring_of&oldid=35072