Properties of an Idempotent

[jacobsldr]

We are given that element A is an idempotent for a ring R. I need to prove a certain set (S={ABA, B element of R}) is a subring. I can go through the subring test no problem, but somehow I need to relate the fact that the idempotent A will also be the unity of S to make it work. My guess is that A will also be the unity of R, but we are not given that R is a ring with unity so I can't assume that. My text doesn't give me much on idempotents except that A^2=A so I am having a hard time making the leap as to why A would have to be the unity of S (and I am pretty sure that it will be 1 for this example).

I know that if A is 1 then of course 1^2=1, but since I wasn't given that R is a ring with unity how can I know that R even contains the element 1?

If anyone could shed some light on how I might proceed, it would be much appreciated.

Thank you.

 

[daon]

If g is an element of S, then g = ABA for some B in R.

Then g*A = ABA*A = ABA^2 = ABA

Similarly, A*g = A*ABA = A^2BA = ABA.

So, for any element g in S, g*A = A*g = g.

 

[jacobsldr]

Excellent, thank you.