Difference between revisions of "Primary ring"
From Encyclopedia of Mathematics
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| − | A [[Ring|ring]] with a unit whose quotient ring with respect to the [[ | + | 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==== | ||
| − | <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> | + | <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> | ||
====Comments==== | ====Comments==== | ||
| − | See also [[ | + | See also [[Nil ideal]]. |
| + | |||
| + | [[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=32821
Primary ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primary_ring&oldid=32821
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article