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Difference between revisions of "Fractions, ring of"

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(→‎Comments: See Field of fractions)
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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
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{{TEX|done}}
 
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A ring related to a given associative ring $R$ with an identity. The (right classical) ring of fractions of $R$ is the ring $Q_{\mathrm{cl}}(R)$ in which every regular element (that is, not a [[zero divisor]]) of $R$ is invertible, and every element of $Q_{\mathrm{cl}}(R)$ has the form $ab^{-1}$ with $a,b \in R$. The ring $Q_{\mathrm{cl}}(R)$ exists if and only if $R$ satisfies the right-hand Ore condition (cf. [[Associative rings and algebras]]). The maximal (or complete) right ring of fractions of $R$ is the ring $Q_{\mathrm{max}}(R) = \mathrm{Hom}_H(\widehat R,\widehat R)$, where $\widehat R$ is the [[injective hull]] of $R$ as a right $R$-module, and $H = \mathrm{Hom}_R(\widehat R,\widehat R)$ is the endomorphism ring of the right $R$-module $\widehat R$. The ring $Q_{\mathrm{max}}(R)$ 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>
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$$
 
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\lim_{\rightarrow} \mathrm{Hom}(D,R)
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
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$$
 
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where $D$ is the set of all dense right ideals of $R$ (a right ideal $D$ of a ring $R$ 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>
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$$
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\forall 0 \neq r_1,r_2 \in R\ \exists r\in R\ (r_1r \neq0,\,r_2r \in D)\ .
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$$
  
 
====References====
 
====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>
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<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>
  
  

Latest revision as of 07:26, 29 November 2014

A ring related to a given associative ring $R$ with an identity. The (right classical) ring of fractions of $R$ is the ring $Q_{\mathrm{cl}}(R)$ in which every regular element (that is, not a zero divisor) of $R$ is invertible, and every element of $Q_{\mathrm{cl}}(R)$ has the form $ab^{-1}$ with $a,b \in R$. The ring $Q_{\mathrm{cl}}(R)$ exists if and only if $R$ satisfies the right-hand Ore condition (cf. Associative rings and algebras). The maximal (or complete) right ring of fractions of $R$ is the ring $Q_{\mathrm{max}}(R) = \mathrm{Hom}_H(\widehat R,\widehat R)$, where $\widehat R$ is the injective hull of $R$ as a right $R$-module, and $H = \mathrm{Hom}_R(\widehat R,\widehat R)$ is the endomorphism ring of the right $R$-module $\widehat R$. The ring $Q_{\mathrm{max}}(R)$ can also be defined as the direct limit $$ \lim_{\rightarrow} \mathrm{Hom}(D,R) $$ where $D$ is the set of all dense right ideals of $R$ (a right ideal $D$ of a ring $R$ is called a dense ideal if $$ \forall 0 \neq r_1,r_2 \in R\ \exists r\in R\ (r_1r \neq0,\,r_2r \in D)\ . $$

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.

How to Cite This Entry:
Fractions, ring of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fractions,_ring_of&oldid=35066
This article was adapted from an original article by L.A. Bokut' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article