
subring
Hi ! i have this exercise , i need help
For a fixed element a $\displaystyle \in{} \mathbb{R}$ we have $\displaystyle C(a)=r \in{} R\ /\ ra=ar$ . Prove that $\displaystyle C(a)$ is a subring of R containing a . Prove that the center of R is the intersection of the subrings $\displaystyle C (a)$ for all s $\displaystyle \in{}$ R

Re: subring
in this case, i believe that you want to show this for an arbitrary ring (possibly noncommutative) R, *not* the ring of real numbers (because every real number centralizes every other real number, because multiplication is commutative on the reals).
you need to show 3 things:
1) if x,y are in C(a), then x  y is in C(a).
2) if x,y are in C(a) , so is xy.
3) C(a) is nonempty (so we can actually use (1) & (2)).
proving a is in C(a) will show (3). does aa = aa?
the distributive law should figure heavily in your proof of (1). the associative law of multiplication should figure heavily in your proof of (2).
to prove that:
$\displaystyle C(R) = \bigcap_{a \in R} C(a)$,
is equivalent to saying that for r in C(R), ra = ar for EVERY a in R. is that not the definition of C(R)?