Thread: Proof-Abstract Algebra

The problem is to prove that the existence of an idempotent (an element whose "square" is itself, that is b * b = b) is a structural property.

I would just like someone to proofread my proof and give me some feedback....Thanx! (note: E="an element of")

Pf: Let <S,*> and <T,#> be isomorphic algebraic structures and suppose a*a=a, where a E S. Assume f: <S,*> --> <T,#> is an isomorphism. Then x=f(a) E T. Now,

x#x = f(a) # f(a) by substitution
= f(a*a) by operation preserving
= f(a) since a*a=a
= x as desired.
Hence, the existence of an idempotent is a srtuctural property.

I feel confident that my proof is OK...but is there anyone that can proofread this and give me any feedback? Any comments are welcome and I take constructive criticizm very well; that's how I learn.