Is there any nontrivial commutative ring without multiplicative identity that satisfies alternating property ($x \cdot x = 0$ for all $x$ where $\cdot$ is multiplication operator and $x \cdot y \neq 0$ when $x \neq y$ and $x$ and $y$ are nonzero except in the case of $x \cdot (x \cdot y)=0$)?
The commutative ring may be a commutative ring that does not have multiplicative identity.
The total cardinality of distinct elements of the ring must be infinite. (So this eliminates the case like $Z/7Z$.)
And what happens if we eliminate the additive inverse axiom from commutative ring mentioned above? I know that it will no longer be a ring, but that is fine. So this would be commutative semiring without requirement of multiplicative identity.
so I am basically asking for two cases: commutative semiring or commutative ring - with both not requiring multiplicative identity.