In mathematics, a semi-local ring is a ring for which R/J(R) is a semisimple ring, where J(R) is the Jacobson radical of R.
The above definition is satisfied if R has a finite number of maximal right ideals (and finite number of maximal left ideals). When R is a commutative ring, the converse implication is also true, and so the definition of semi-local for commutative rings is often taken to be "having finitely many maximal ideals".
Some literature refers to a commutative semi-local ring in general as aquasi-semi-local ring, using semi-local ring to refer to a Noetherian ring with finitely many maximal ideals.
A semi-local ring is thus more general than a local ring, which has only one maximal (right/left/two-sided) ideal.
Z/mZ
m
Z/mZ
n{F | |
oplus | |
i} |
n | |
R/cap | |
i=1 |
mi\congoplus
n | |
i=1 |
R/mi
(The map is the natural projection). The right hand side is a direct sum of fields. Here we note that ∩i mi=J(R), and we see that R/J(R) is indeed a semisimple ring.