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.
A set R with two binary operations + and · such that (R,+) is an abelian group with identity 0, and
a
(
b
+
c
)
+
a
0
=
a
b
+
a
c
and
(
b
+
c
)
a
+
0
a
=
b
a
+
c
a
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.
