# Primary ring

From Encyclopedia of Mathematics

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=33710

This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article