cardinality of elements in a "semiring minus multiplicative identity"

Question

In a theory that has all axioms of semiring except multiplicative identity axiom, will there be a model of the theory that has infinite elements? The model must violate multiplicative identity axiom.

Answer 1

Sure, take $2 \mathbb{Z}$ or $2 \mathbb{N}$. With products and other standard constructions you can construct as many examples as you want.

Answer 2

You can pick any infinite ring.