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, denoted C_R(S) in this article. It is always the case that C_R(C_R(S)) contains S, and a double centralizer theorem gives conditions on R and S that guarantee that C_R(C_R(S)) is equal to S.

The centralizer of a subring S of R given by

Clearly C_R(C_R(S)) ⊇ S, but it is not always the case that one can say the two sets are equal. The double centralizer theorems give conditions under which one can conclude that equality occurs.

There is another special case of interest. Let M be a right R module and give M the natural left E-module structure, where E is End(M), the ring of 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 → m_r is a ring homomorphism of R into the ring E, and we denote the image of R inside of E by R_M. It can be checked that the kernel of this canonical map is the annihilator Ann(M_R). Therefore, by an isomorphism theorem for rings, R_M is isomorphic to the quotient ring R/Ann(M_R). Clearly when M is a faithful module, R and R_M are isomorphic rings.

So now E is a ring with R_M as a subring, and C_E(R_M) may be formed. By definition one can check that C_E(R_M) = End(M_R), the ring of R module endomorphisms of M. Thus if it occurs that C_E(C_E(R_M)) = R_M, this is the same thing as saying C_E(End(M_R)) = R_M.

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