What does Herstein mean by 'centroid of a ring'?

Question

I'm currently reading Herstein's Noncommutative Rings, and the definition of the centroid of a ring is on page 46 of the book.

Let $\text{End}(R)$ be the ring of endomorphisms of the additive group $R$. For each $a \in R$, define $$\begin{array}{rccc}T_a: & R & \longrightarrow & R \\ & x & \longmapsto & xa \end{array}$$ and $$\begin{array}{rccc}L_a: & R & \longrightarrow & R \\ & x & \longmapsto & ax \end{array}.$$

Now, let $\text{B}(R)$ be the subring of $\text{End}(R)$ generated by all elements $L_a$, and $T_a$.

Centroid definition

The centroid of $R$ is the set of elements in $\text{End}(R)$ which commute element-wise with $\text{B}(R)$.

Since I'm Googling for some examples on this definition, but I found none (even wikipedia doesn't have it), so I guess this terminology is obsolete. Is there any newer terminology for the centroid of a ring?

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Edit (28, July, 2013)

Thanks to user85319's suggestion of the book. So I'm posting here an extract of the book Fields and Rings by Irving Kaplansky.

I do have some questions left. First, on page 147, he gives the definition as follow:

Page 147

let $A$ be any ring, associated with an element $x \in A$, we have the right, and left multiplication:

$$R(x):a \mapsto ax$$ $$L(x):a \mapsto xa$$

The ring $E$ generated by all the $L$'s, and $R$'s are called the envelop of $A$ (which I take to be the same as $\text{B}(R)$ in Herstein's definition). It is the subring of the ring of all endomorphism of the Abelian group $A$. The general element of $E$ is the sum of terms, each of which is the product of $L$'s, and $R$'s.

When $E$ is assiciative, the general term of the element in $E$ takes on the simplier form:

$$L(a) + R(b) + \sum L(c_i)R(d_i)$$

We think of $E$ as placed on the right of $A$, and in this way $A$ becomes a right $E-$module. Examining the relevant definition, we see that $A$ is simple iff it's an irreducible $E-$module.

What I don't understand are the two bolded parts above. I have no idea about the second part, but I have some on the first part.

What I'm not sure is why $E$ needs to be associative? Say I have: $xL({a_1})R({b_1})L({a_2})R({b_2}) = a_2a_1xb_1b_2 = xL({a_2a_1})R({b_2b_1})$, i.e, I can kind of combining many of the left, and right multiplications into one pair of $L(c)R(d)$, and I can do the same to the other terms without using the fact that $E$ is associative. Am I missing something?

Thank you guys very much,

And have a good day.

Answer 1

Centroids (and extended centroids) play a role in the theory of (generalized) polynomial identities and functional identities in rings. One of the more recent books in this direction is Functional Identities by Matej Bresar, Mikhail A. Chebotar, Wallace S. Martindale.

Answer 2

Ok, after thinking for a while (actually, for more than a day), is it true that the centroid of some ring $R$ is the set of all endomorphisms of the $R-$bimodule $R$?

I think it is, because: Let $\tau \in \text{C}(R)$, where $\text{C}(R)$ denotes the centroid of $R$. Then we'll have:

1. $\tau(a + b) = \tau(a) + \tau(b)$, since we can view $\tau$ to be some additive group endomorphism of $R$.

2. $\tau(a.b) = \tau(T_b(a)) = T_b(\tau(a)) = \tau(a).b$.

3. $\tau(a.b) = \tau(L_a(b)) = L_a(\tau(b)) = a.\tau(b)$.

Is it true? Or am I missing something here?