- 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 (R,+) is an abelian group with identity 0, and and for all a, b, c in R.
- An abelian group (A,+) equipped with a subgroup B and a multiplication B × A → A making B a ring and A a B-module.
No two of these definitions are equivalent, so it is best to avoid the term "pseudo-ring" or to clarify which meaning is intended.
- Semiring – an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse