Semilocal ring

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. (Lam 2001, p. §20)(Mikhalev & Pilz 2002, p. C.7)

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 a quasi-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.

Examples

• Any right or left Artinian ring, any serial ring, and any semiperfect ring is semi-local.
• The quotient Z / m Z {\displaystyle \mathbb {Z} /m\mathbb {Z} } is a semi-local ring. In particular, if m {\displaystyle m} is a prime power, then Z / m Z {\displaystyle \mathbb {Z} /m\mathbb {Z} } is a local ring.
• A finite direct sum of fields ⨁ i = 1 n F i {\displaystyle \bigoplus _{i=1}^{n}{F_{i}}} is a semi-local ring.
• In the case of commutative rings with unity, this example is prototypical in the following sense: the Chinese remainder theorem shows that for a semi-local commutative ring R with unit and maximal ideals m₁, ..., m_n

R / ⋂ i = 1 n m i ≅ ⨁ i = 1 n R / m i {\displaystyle R/\bigcap _{i=1}^{n}m_{i}\cong \bigoplus _{i=1}^{n}R/m_{i}\,} .

(The map is the natural projection). The right hand side is a direct sum of fields. Here we note that ∩_i m_i=J(R), and we see that R/J(R) is indeed a semisimple ring.

• The classical ring of quotients for any commutative Noetherian ring is a semilocal ring.
• The endomorphism ring of an Artinian module is a semilocal ring.
• Semi-local rings occur for example in algebraic geometry when a (commutative) ring R is localized with respect to the multiplicatively closed subset S = ∩ (R \ p_i), where the p_i are finitely many prime ideals.

References