Double centralizer theorem

In the branch of abstract algebra called ring theory, the Double centralizer theorem can refer to any one of several similar results. These results concern the centralizer of a subring S of a ring R, which will be denoted $\mathrm {C} _{R}(S)\,$ in this article. In short, it is always the case that $\mathrm {C} _{R}(\mathrm {C} _{R}(S))\,$ contains S, and a Double centralizer theorem gives conditions on R and S which guarantee that $\mathrm {C} _{R}(\mathrm {C} _{R}(S))\,$ is equal to S.

Statements of the theorem

Central simple algebras

Perhaps the most common version is the version for central simple algebras, as it appears in (Knapp 2007, p.115):

Theorem: If A is a finite dimensional central simple algebra over a field F and B is a simple subalgebra of A, then $\mathrm {C} _{A}(\mathrm {C} _{A}(B))=B\,$, and moreover the dimensions satisfy $\mathrm {dim} _{F}(B)\cdot \mathrm {dim} _{F}(\mathrm {C} _{A}(B))=\mathrm {dim} _{F}(A)\,$.

To be absolutely clear, if S is a subring of R, the notation for the centralizer is

$C_{R}(S):=\{r\in R\mid rs=sr{\text{ for all }}s\in S\}$.

Artinian rings

The following generalized version for Artinian rings (which include finite dimensional algebras) appears in (Isaacs 2009, p.187). A few preliminaries are necessary to state this version.

(MOVE TO MOTIVATION SECTION) Let M be a right R module and give M the natural left E module structure where E=End(M) (the ring of R endomorphisms of the abelian group M). Every map m_r given by m_r(x)=xr creates an additive endomorphism of M, that is, an element of E. The map $r\mapsto m_{r}$ is a ring homomorphism of R into the endomorphism ring $E=\mathrm {End} (M)\,$, and we denote the image of R in E by R_U.

By definition, the ring of R-endomorphisms of M is

$\mathrm {End} (M_{R}):=\{f\in E\mid f(mr)=f(m)r{\text{ for all }}x\in M,r\in R\}=C_{E}(R_{M})$

For general modules, there may be more elements of End(_EM) than those arising this way.

Firstly, given a module M_R and an element r from R, the map m_r given by $u\mapsto ur$ is an endomorphism of the additive abelian group U.

The ring $D=\mathrm {End} (U_{R})\,$ of R endomorphisms of U is a subring of E, and is a division ring by Schur's lemma. It turns out that R_U ⊆D.

Theorem: If R is a right Artinian ring and U_R is a simple right module and R_U, D and E are given as in the previous paragraph, then

$R_{U}=\mathrm {C} _{E}(\mathrm {C} _{E}(R_{U}))\,$.

Remarks:

• Since algebras are normally defined over commutative rings, and all the involved rings above may be noncommutative, it's clear that algebras are not necessarily involved.
• If U is additionally a faithful module, so that R is a right primitive ring, then R_U is ring isomorphic to R.
• In this version, the rings are chosen with the intent of proving the Jacobson density theorem.

Polynomial identity rings

In (Rowen 1980, p.154), a version is given for polynomial identity rings. The notation Cen(R) will be used to denote the center of a ring R.

Theorem: If R is a simple polynomial identity ring, and A is a simple Cen(R) subalgebra of R, then $\mathrm {C} _{A}(\mathrm {C} _{R}(A))=A\,$.

Remarks:

• This version generalizes the F algebra version by considering R as a Cen(R) algebra, but in this version R need not be right Artinian. For example, all commutative non-Artinian rings, and all matrix ring over such rings are all non-Artinian polynomial identity rings.

Double centralizer property

Main article: Balanced module

A module M is said to have the double centralizer property or to be a balanced module if every E endomorphism of M is given by right multiplication by some element r of R. Explicitly, if f is in End(_EM) then there exists an r in R such that f(x)=xr for all x in M.

References

• Isaacs, I. Martin (2009), Algebra: a graduate course, Graduate Studies in Mathematics, vol. 100, Providence, RI: American Mathematical Society, pp. xii+516, ISBN 978-0-8218-4799-2, MR 2472787 {{citation}}: Unknown parameter |note= ignored (help)

• Knapp, Anthony W. (2007), Advanced algebra, Cornerstones, Boston, MA: Birkhäuser Boston Inc., pp. xxiv+730, ISBN 978-0-8176-4522-9, MR 2360434

• Rowen, Louis Halle (1980), Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, New York: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], pp. xx+365, ISBN 0-12-599850-3, MR 0576061