Pseudo-ring explained

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 is an abelian group with identity 0, and and for all a, b, c in R.^[2]
• An abelian group equipped with a subgroup B and a multiplication making B a ring and A a B-module.^[3]

None 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

Notes and References

1. Book: Bourbaki . N. . Nicolas Bourbaki . Algebra I, Chapters 1-3 . Springer . 1998 . 98.
2. Natarajan. N. S.. Rings with generalised distributive laws. J. Indian. Math. Soc. . New Series. 1964. 28. 1–6.
3. Patterson. Edward M.. The Jacobson radical of a pseudo-ring. Math. Z.. 1965. 89. 4. 348–364. 10.1007/bf01112167. 120796340.