Proving Z is a Ring with Addition and Subtraction

[lockedup]

Homework Statement

Prove that Z with the following addition and subtraction is a ring.

Homework Equations

a[tex]\oplus[/tex]b = a + b - 1 and a[tex]\odot[/tex]b = ab - (a + b) + 2

The Attempt at a Solution

I proved all the axioms for addition. I'm stuck on the multiplication part.

(a[tex]\odot[/tex]b)[tex]\odot[/tex]c = (ab-(a+b)+2)c - (ab-(a+b)+2+c) + 2

a[tex]\odot[/tex](b[tex]\odot[/tex]c) = a(bc-(b+c)+2) - (a+bc-(b+c)+2) + 2

How are these equal? I know it's a ring because a couple problems later, my books wants me to prove that it's an integral domain...

 

[Dick]

I think they are equal. Just expand them out.

 

[lockedup]

(ab-(a+b)+2)c - (ab-(a+b)+2+c) + 2 =

abc-ac-bc+2c-ab+a+b-2-c+2 = abc-ac-bc-ab+c+a+b

a(bc-(b+c)+2) - (a+bc-(b+c)+2) + 2 =

abc-ab-ac+2a-a-bc+b+c-2+2 = abc-ab-ac-bc+a+b+c

It looks so much clearer now. My own handwriting was deceiving me... Gah, that's the second stupid question I've posted this weekend...