Let R be a ring that contains at least two elements. Suppose for each nonzero a in R there exists a unique b in R such that aba = a.

a. show that R has no divisors of 0

b. show that bab = b

c. show that R has unity

d. show that R is a division ring

B I worked out but I'm not sure if it works..

we're given aba = a and we need bab=b

abab= ab

multiply by inverse of a to get bab=b

thanks!