Basic properties of centralizers in a ring

Let R be a ring. For a fixed element a \in R, define C_R(a) = \{r \in R \ |\ ra = ar \}. Prove that C_R(a) is a subring of R containing a. Prove that Z(R) = \bigcap_{a \in R} C_R(a).


(We need not assume that R has a 1.)

Note that aa = aa, so that a \in C_R(a). Similarly, 0a = a0, so 0 \in C_R(a). If R has a 1, then since 1a = a1, 1 \in C_R(a).

Now if x,y \in C_R(a), then (x-y)a = xa-ya = ax-ay = a(x-y), so that x-y \in C_R(a), and xya = xay = axy, so that xy \in C_R(a). Thus C_R(a) is a subring.

If x \in Z(R), then for all r \in R, xr = rx. Thus for all r \in R, x \in C_R(r), and we have x \in \bigcap_{r \in R} C_R(r).

Now if x \in \bigcap_{r \in R} C_R(r), then for all r \in R, x \in C_R(r). So xr = rx, and we have x \in Z(R).

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