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Difference between revisions of "Primary ring"

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A [[Ring|ring]] with a unit whose quotient ring with respect to the [[Jacobson radical|Jacobson radical]] is isomorphic to a matrix ring over a skew field, or, which is the same, is an Artinian simple ring. If the idempotents of a primary ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074480/p0744801.png" /> with Jacobson radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074480/p0744802.png" /> can be lifted modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074480/p0744803.png" /> (i.e. for every idempotent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074480/p0744804.png" /> there is an idempotent pre-image in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074480/p0744805.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074480/p0744806.png" /> is isomorphic to the full matrix ring of a local ring. This holds, in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074480/p0744807.png" /> is a nil ideal.
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A [[Ring|ring]] with a unit whose quotient ring with respect to the [[Jacobson radical]] is isomorphic to a [[matrix ring]] over a [[skew-field]], or, which is the same, is an Artinian simple ring (cf [[Artinian ring]], [[Simple ring]]). If the [[idempotent]]s of a primary ring $R$ with Jacobson radical $J$ can be lifted modulo $J$ (i.e. for every idempotent of $R/J$ there is an idempotent pre-image in $R$), then $R$ is isomorphic to the full matrix ring of a [[local ring]]. This holds, in particular, if $J$ is a [[nil ideal]].
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Faith,  "Algebra" , '''1–2''' , Springer  (1973–1976)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  C. Faith,  "Algebra" , '''1–2''' , Springer  (1973–1976)</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
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See also [[Nil ideal|Nil ideal]].
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See also [[Nil ideal]].
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[[Category:Associative rings and algebras]]

Latest revision as of 16:49, 17 October 2014

A ring with a unit whose quotient ring with respect to the Jacobson radical is isomorphic to a matrix ring over a skew-field, or, which is the same, is an Artinian simple ring (cf Artinian ring, Simple ring). If the idempotents of a primary ring $R$ with Jacobson radical $J$ can be lifted modulo $J$ (i.e. for every idempotent of $R/J$ there is an idempotent pre-image in $R$), then $R$ is isomorphic to the full matrix ring of a local ring. This holds, in particular, if $J$ is a nil ideal.

References

[1] N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
[2] C. Faith, "Algebra" , 1–2 , Springer (1973–1976)


Comments

See also Nil ideal.

How to Cite This Entry:
Primary ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primary_ring&oldid=17882
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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