96 2 Algebra
Using the fact thaty^4 =y^2 , we see that this is equal to zero, and hencexy^2 −y^2 xy^2 =0,
that is,xy^2 =y^2 xy^2. A similar argument shows thaty^2 x=y^2 xy^2 , and soxy^2 =y^2 x
for allx, y∈R.
Using this we obtain
xy=xyxyxy=xy(xy)^2 =x(xy)^2 y=x^2 yxy^2 =y^3 x^3 =yx.This proves that the ring is commutative, as desired.
We remark that both this and the second problem below are particular cases of a
general theorem of Jacobson, which states that if a ring (with or without identity) has the
property that for every elementxthere exists an integern(x) >1 such thatxn(x)=x,
then the ring is commutative.
291.LetRbe a nontrivial ring with identity, andM={x∈R|x=x^2 }the set of its
idempotents. Prove that ifMis finite, then it has an even number of elements.
292.LetRbe a ring with identity such thatx^6 =xfor allx∈R. Prove thatx^2 =xfor
allx∈R. Prove that any such ring is commutative.
293.LetRbe a ring with identity with the property that(xy)^2 =x^2 y^2 for allx, y∈R.
Show thatRis commutative.
294.Letxandybe elements in a ring with identity andna positive integer. Prove that
if 1−(xy)nis invertible, then so is 1−(yx)n.
295.LetRbe a ring with the property that ifx∈Randx^2 =0, thenx=0.
(a) Prove that ifx, z∈Randz^2 =z, thenzxz−xz=0.
(b) Prove that any idempotent ofRbelongs to the center ofR(the center of a ring
consists of those elements that commute with all elements of the ring).
296.Show that if a ringRwith identity has three elementsa, b, csuch that
(i)ab=ba,bc=cb;
(ii) for anyx, y∈R,bx=byimpliesx=y;
(iii)ca=bbutac =b,
then the ring cannot be finite.