---
tag: ZDCDS6EWGX
kind: concept
name: Semilocal ring
msc: [16L30]
sources:
  - url: https://en.wikipedia.org/wiki/Semi-local_ring
---

## Summary

A semilocal ring is a ring $R$ whose quotient $R/J(R)$ by the [[Jacobson radical|XA0DCNH7WY]] is a [[semisimple ring|NY25HXA478]].
For a [[commutative ring|0310REJJY8]] this is the same as having finitely many maximal ideals, the [[commutative notion|6DYAGW68AV]].
Left Artinian rings and [[local rings|MZ02BKPHJA]] are semilocal.
Semilocal rings are the starting point for [[semiperfect|ZRC2S3ECFK]], [[perfect|49Y9Q949PK]] and [[semiprimary|XVP517QRH5]] rings, which add conditions on $J(R)$.

## Explanations

### L1 human

Dividing a ring by its Jacobson radical removes the part that simple modules cannot see.
For a semilocal ring, what remains is as simple as possible: a finite product of matrix rings over division rings.
For example, the ring of upper triangular $2 \times 2$ matrices over a field is semilocal, since modulo its radical it is a product of two copies of the field.
The integers are not semilocal, since they have infinitely many maximal ideals.