Primary ring
From Encyclopedia of Mathematics
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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. If the idempotents of a primary ring 
with Jacobson radical 
can be lifted modulo 
(i.e. for every idempotent of 
there is an idempotent pre-image in 
), then 
is isomorphic to the full matrix ring of a local ring. This holds, in particular, if 
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
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