Prove for all integers $x$ and $y$: $x y = x$ → $y = 1$ using field axioms.

Question

Knowing for all integers $x$ and $y$: $x y = x$, prove $y = 1$ using only the following axioms: Associativity of addition. Existence of additive identity. Existence of additive inverses. Commutativity of multiplication. Associativity of multiplication. Existence of multiplicative identity. Distributive law.

Note that there's no multiplication inverse, is this proof still possible?

Answer 1

Let $x = 1 \implies 1\cdot y = 1 \implies y = 1$