# Deﬁne a new addition ⊕ and multiplication on Z by a⊕b = a + b−1 and ab = a + b−ab.

a+b and ab are the usual integer addition and multiplication. You can assume that this new operation forms a ring, say R is the set of integers with these operations. Then does R have zero-divisors? And Are R and Z (the integers) isomorphic?

Any help will be appreciated!

• It is a terrible idea to split the question between title and body. Write the entire question in the body please. – rschwieb May 2 '16 at 23:08

There's a simple isomorphism $$\varphi : R \to \mathbb{Z},~~z \mapsto 1-z$$ From this it immediately follows that $R$ does not have zero divisors, since $\mathbb{Z}$ does not either.