I'm a little bit stuck on an assignment question.

The question is:

I already proved that (Quote:

Letcbe a fixed positive integer, and let * denote the binary operation on the setZof integers defined be the formula

$\displaystyle x * y = xy + c(x+y) + c^2 - c$

for all integers x, y, and z.

Is (Z, *) a monoid? If so, what is it's identity element?

Z, *) is a semigroup. I also showed that x*e = e*x . I just don't know how to find e. (That is, if e exists at all)

This is what I have so far:

$\displaystyle x*e = xe + c(x+e) + c^2 - c$

$\displaystyle

e*x = ex + c(e+x) + c^2 - c

$

Any help would be much appreciated.

EDIT: I'm pretty sure it's NOT a monoid. Do you think I need to prove this though? Can I just say something along the lines of "clearly, there is no identity element" :P

$\displaystyle x * e = xe + c(x+e) + c^2 - c \neq x$