# 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.[1]
- 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*.[2] - An abelian group (
*A*,+) equipped with a subgroup*B*and a multiplication*B*×*A*→*A*making*B*a ring and*A*a*B*-module.[3]

No two of these definitions are equivalent, so it is best to avoid the term "pseudo-ring" or to clarify which meaning is intended.

## See also

- Semiring – an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse

## References

- Bourbaki, N. (1998).
*Algebra I, Chapters 1-3*. Springer. p. 98. - Natarajan, N. S. (1964). "Rings with generalised distributive laws".
*J. Indian. Math. Soc. (N. S.)*.**28**: 1–6. - Patterson, Edward M. (1965). "The Jacobson radical of a pseudo-ring".
*Math. Z*.**89**: 348–364. doi:10.1007/bf01112167.

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