This is for abstract algebra class. Please show me all the details from scratch please. Thank you.

6. Let R be a ring with more than one element such that for each nonzero a R, there exists a unique b R such that aba = a. (a) Prove that R has no zero divisors. (b) Let a be a nonzero element of R and let b be the unique element of R such that aba = a. Prove that bab = b. () Prove that R has a multiplicative identity. (d) Prove that R is a division ring