Let R middot be a ring such that R is a cyclic group Prove

Let (R, +, middot) be a ring such that (R, +) is a cyclic group. Prove that R is a commutative ring.

Solution

(R,+) is given to be a cyclic group.

To show that R is a commutative ring.

As R is a cyclic group, we may identify it with either Z or Z_n. (integers modulo n), generated by 1.

Any element x of R is of the form k.1 where k is an integer.

We may assume k positive (by changing x to -x if necessary).

So let x=k.1 and y =m.1.

Then xy = (1+1+...+1).(1+1+...+1) (the first factor has k terms, the second m terms)

              =km .1 ( from axioms of Rings)

similarly yx =mk.1

as km =mk in integers , it follows that xy=yx for all x,y in R.

Hence R is commutative.