I have some true or false questions and would like to have your help to check on it.
A). in a ring R, if $x^2=x$, $\forall x\in R$, then R is commutative
For (A), when looking at $(x+y)^2$, it has $x+y=(x+y)^2=x^2+xy+yx+y^2$
and then yx+xy=0, and from 2x=4x, therefore 2x=0. how this play a role here?
B) In an integral domain, it $\exists m\in N, s,t, mx=0, \forall x\in R$, then it's a finite integral domain.
I think this one is correct by definition of characteristic for a ring/field.
C) commutative ring with unity has at least two elements, and cancellation holds, then it's an integral domain.
I think it's correct,
first it's a commutative ring with unity,
second, it has at least two element, and there's a property saying that an integral domain must have at least two elements,
and the third, cancellation holds implies it has no zero divisor.
So the three above looks like fit the profile of integral domain.
D) f is a homomorphism from group G to group H,
then f(G) is a normal subgroup in H
I'm not quite sure about this one, kind of remember that normal subg roup of G under homomorphism is normal subgroup of H, but don't know if f(G) will be normal in H
Are these answer or argument correct? Thanks for your help.