Pseudo-ring

In mathematics, and more specifically in abstract algebra, a pseudo-ring is one of the following variants of a ring:

A rng, i.e., a structure satisfying all the axioms of a ring except for the existence of a multiplicative identity.
A set R with two binary operations + and ⋅ such that is an abelian group with identity 0, and and for all a, b, c in R.
An abelian group equipped with a subgroup B and a multiplication making B a ring and A a B-module.

None of these definitions are equivalent, so it is best to avoid the term "pseudo-ring" or to clarify which meaning is intended.