Q: is a ring with following property
1. It has a unit element, 1.
2.
Prove R is commutative.
(Source: Herstein - Problem#22 Pg 168 Ch 3 Ring Theory)
I did it in a very round about way. First proving bab = abb, aba=aab and then the final result ab = ba.
I essentially used
(a+1)(b+1)(a+1)(b+1) = (a+1)(a+1)(b+1)(b+1)
holds true for this ring.
Not too happy with my attempt. I think I just got the final result by a fluke. Is there any better/quick method / structured approach for this problem.
problem is that when a ring R has unit element 1 it doesn't mean that all elements of R are units (i.e. they all have multiplicative inverses) - consider ring for example.
i like the way how aman cc proved commutativity, it is correct (supposing he got bab = abb from identity (1+a)b(1+a)b = (1+a)(1+a)bb etc.). i don't know simpler proof.