Why called double centralizer property?

Question

This is related to Lam A first course in non-commutative rings Thm 3.11 of Chpt 1, Sec 3.

Thm 3.11 Let $R$ be a simple ring, $0\neq A\subset R$ is a left ideal. Let $D=End(_RA)$ (R-linear maps) which acts on $A$ on the right side. Then the natural map $f:R\to End(A_D)$ (D-linear maps) is isomorphism by $r\to r\cdot$.

Then the book comments this property of $ _RA$ is referred as double centralizer property.

$\textbf{Q:}$ Why is this called double centralizer?

Answer 1

Let $S$ be the group of all endomorphisms of $A$ as an abelian group. We can identify $R$ with a subring of $S$ via the left action on $A$. Then $D$ is exactly the centralizer of $R$ inside $S$, and $\operatorname{End}(A_D)$ is exactly the centralizer of $D$ inside $S$. So to say that $f$ is an isomorphism is the same as saying that $R$ is equal to its own double centralizer, as a subset of $S$.