So here's the problem: In each of the following, a set A w/ operations addition and multiplication is given. Prove that A satisfies all the axioms to be commutative ring w/unity. Indicate the zero element, the unity and the negative of an arbitrary a.
1.) A is the zet of Z of integers with the following "addition" + and "multiplication" x:
a+b=a+b+1 ; axb=ab-(a+b)+2.
So i worked it out but my teacher said i need to show the distributive property?
do i have to multiply by an x, like x(a+b+1) and x((a+b)+2)?
denote your version of addition and
denote your version of multiplication so that
Then you need to show that
and if you have shown commutativity of
that's enough, if not, you also need to show