A nonzero element $a$ in a commutative ring $R$ is called a zero divisor if there is a non zero element $b\in R$ such that $ab=0$.
Consider the set $\mathbb Z$ with the operations $\oplus$ and $\otimes$ defined for $a, b \in \mathbb Z$ by $a\oplus b=a+b−1$ and $a \otimes b = ab − (a + b) + 2$.
Are there any zero divisors in this ring?
Do we need to check zero divisor in case of non-commutative ring.
The solution I have for zero divisor part is
Solution: No, there are no zero divisors in this ring. It is crucial to remember that the “zero element” in this ring is the additive identity, namely, 1. So, we need to check whether there are elements a = 1 and b = 1 such that a ⊗ b = 1, i.e. ab − (a + b) + 2 = 1 (a − 1)(b − 1) = 0 The only way for this equation to hold is that either a = 1 or b = 1. Thus there can be no zero divisors in this ring.
Why we are checking a ⊗ b = 1 ,we should check a ⊗ b = 0 where $a,b \neq 0$