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 ${\displaystyle a(b+c)+a0=ab+ac}$ and ${\displaystyle (b+c)a+0a=ba+ca}$ 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

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