# 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 × AA 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.