Semilocal rings

Posted: August 5, 2022 in Local & Semilocal Rings, Noncommutative Ring Theory Notes
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All rings in this post are assumed to have 1 and they may or may not be commutative. As always, for a ring R, we denote by J(R) the Jacobson radical of R.

Here we defined a local ring as a ring with only one maximal left ideal. That was not a strange definition since we already knew the definition of a commutative local ring: a commutative ring is said to be local if it has only one maximal ideal.

Now we want to define semilocal rings. Well, we already know the definition for commutative rings: a commutative ring is said to be semilocal if it has only finitely many maximal ideals. However, if R is not necessarily commutative, things get a little bit weird since algebraists do not seem to agree on the definition of semilocal rings. One thing they seem to agree on is that semilocal rings should not be defined as expected, i.e. rings with only finitely many maximal left ideals. However, algebraists want the class of semilocal rings to contain rings with only finitely many maximal left ideals. So semilocal rings are not a straightforward extension of local rings. In this post, I am going to follow T. Y. Lam’s definition of semilocal rings in his book A First Course in Noncommutative Rings.

Definition. We say that a ring R is semilocal if R/J(R) is a left Artinian ring.

Remark 1. In Theorem 1 this post, we showed that R is local if and only if R/J(R) is a division ring. Since division rings are obviously left Artinian, it follows that every local ring is semilocal. So the definition is not too weird.

Remark 2. Since a semiprimitive ring is semisimple if and only if it is left Artinian, and R/J(R) is clearly semiprimitive for any ring R, we conclude that R is semilocal if and only if R/J(R) is semisimple. In other words, by the Artin-Wedderburn theorem, R is semilocal if and only if \displaystyle R/J(R) \cong \prod_{i=1}^kM_{n_i}(D_i), for some positive integers k, n_i with 1 \le i \le k, and some division rings D_i.

The first two parts of the following theorem shows that the definition of semilocal rings is actually not weird at all.

Theorem. Let R be a ring.

i) If R has only finitely many left maximal ideals, then R is semilocal.

ii) If R is commutative, then R is semilocal if and only if R has only finitely many maximal ideals.

iii) If R_1, \cdots , R_n are rings, then R:=\bigoplus_{i=1}^nR_i is semilocal if and only if each R_i is semilocal.

Proof. i) Let \mathfrak{m}_i, \ 1 \le i \le k, be all the maximal left ideals of R. So J(R)=\bigcap_{i=1}^k \mathfrak{m}_i. Consider the map f: R/J(R) \to \bigoplus_{i=1}^k R/\mathfrak{m}_i defined by f(r+J(R))=(r+\mathfrak{m}_1, \cdots , r+ \mathfrak{m}_k), for all r \in R. It is clear that f is a well-defined injective R-module homomorphism. Also, each R/\mathfrak{m}_i is a simple R-module, because each \mathfrak{m}_i is a maximal left ideal of R. In particular, each R/\mathfrak{m}_i is an Artinian R-module and so \bigoplus_{i=1}^k R/\mathfrak{m}_i is an Artinian R-module. Thus every R-submodule of \bigoplus_{i=1}^k R/\mathfrak{m}_i is Artinian, and hence R/J(R) is an Artinian R-module, hence a left Artinian ring.

ii) One side is clear by i). Suppose now that R is a commutative semilocal ring. So, by definition, R/J(R) is a commutative Artinian ring. We showed here (Fact 2) that commutative Artinian rings have only finitely many maximal ideals. So R/J(R) has only finitely many maximal ideals. Since every maximal ideal of R contains J(R), it follows that R has only finitely many maximal ideals.

iii) We have J(R)=\bigoplus_{i=1}^nJ(R_i) and so

R/J(R)=\bigoplus_{i=1}^n R_i/\bigoplus_{i=1}^nJ(R_i) \cong \bigoplus_{i=1}^nR_i/J(R_i).

The result now follows because a finite direct some of rings is left Artinian if and only if each ring is left Artinian (see Exercise 1). \ \Box

Remark 3. Look at the proof of the second part of the above Theorem and see that we really didn’t need R to be commutative, we only needed R/J(R) to be commutative.

Exercise 1. Let R_1, \cdots , R_n be rings and let R:=\bigoplus_{i=1}^n R_i. Show that R is left Artinian (respectively, Noetherian) if and only if each R_i is left Artinian (respectively, Noetherian).
Hint. Show that a left ideal of \bigoplus_{i=1}^n R_i is of the form \bigoplus_{i=1}^n I_i where each I_i is a left ideal of R_i.

Exercise 2. Let R_1, \cdots , R_n be local rings and let R:=\bigoplus_{i=1}^n R_i. Since every local ring is obviously semilocal, R is semilocal, by the above Theorem. Show that R is local if and only if n=1.

Exercise 3. Regarding Remark 3, give an example of a noncommutative semilocal ring R such that R/J(R) is commutative.
Hint. You may choose R to be the ring of 2 \times 2 upper triangular matrices with entries from some field F. Show that J(R)=\left\{\begin{pmatrix}0 & a \\ 0 & 0\end{pmatrix}: \ a \in F\right\} and conclude that R/J(R) \cong F \oplus F.

Exercise 4. Regarding the first part of the Theorem, can you find an example of a semilocal ring with infinitely many maximal left ideals?

Exercise 5. Show that every semilocal ring is Dedekind-finite.
Hint. See Examples 3 and 8 in this post.

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