Thread: Rings, and zero divisors

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!

Originally Posted by lttlbbygurl

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.

suppose but and so there exists a unique such that but then: therefore and hence contradiction.

b. show that bab = b.

we have and thus since the first part of the problem gives us:

c. show that R has unity.

let so there exists such that the claim is that so let then and hence therefore by the first part of the problem.

also by the second part, and thus which gives us thus

d. show that R is a division ring.

let so there exists such that and thus which gives us: