View source for Fractions, ring of

A ring related to a given associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f0413001.png" /> with an identity. The (right classical) ring of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f0413002.png" /> is the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f0413003.png" /> in which every regular element (that is, not a zero divisor) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f0413004.png" /> is invertible, and every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f0413005.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f0413006.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f0413007.png" />. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f0413008.png" /> exists if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f0413009.png" /> satisfies the right-hand Ore condition (cf. [[Associative rings and algebras|Associative rings and algebras]]). The maximal (or complete) right ring of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f04130010.png" /> is the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f04130011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f04130012.png" /> is the injective hull of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f04130013.png" /> as a right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f04130014.png" />-module, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f04130015.png" /> is the endomorphism ring of the right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f04130016.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f04130017.png" />. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f04130018.png" /> can also be defined as the direct limit

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f04130019.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f04130020.png" /> is the set of all dense right ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f04130021.png" /> (a right ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f04130022.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f04130023.png" /> is called a dense ideal if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041300/f04130024.png" /></td> </tr></table>

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



====Comments====
This notion is also called a ring of quotients.