Is a Subring's Unit Element a Zero Divisor if the Ring Lacks a Unit?

[sunjin09]

If the subring S of a ring R has a unit element e' but R does not have a unit element then e' must be a divisor of zero.

I was able to show that if R has a unit element e≠e', then (e-e')e'=0, where e-e'≠0, implying e' is a divisor of zero, but if R does not have a unit element I can't see why, please help, thank you.

 

[Dick]

If the subring S of a ring R has a unit element e' but R does not have a unit element then e' must be a divisor of zero.

I was able to show that if R has a unit element e≠e', then (e-e')e'=0, where e-e'≠0, implying e' is a divisor of zero, but if R does not have a unit element I can't see why, please help, thank you.

If e' isn't a unit in R then there is an element of r of R such that e'r-r is not zero. Multiply that by e'.