- A rng, i.e., a structure satisfying all the axioms of a ring except for that of 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 a(b + c) + a0 = ab + ac and (b + c)a + 0a = ba + ca 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
- Bourbaki, N. (1998). Algebra I, Chapters 1-3. Springer. p. 98.
- Natarajan, N. S. (1964). "Rings with generalised distributive laws". J. Indian. Math. Soc. New Series. 28: 1–6.
- Patterson, Edward M. (1965). "The Jacobson radical of a pseudo-ring". Math. Z. 89 (4): 348–364. doi:10.1007/bf01112167. S2CID 120796340.