Pseudo-ring#Examples

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.{{Cite book

| last1=Bourbaki

| first1=N.

| author1-link=Nicolas Bourbaki

| title=Algebra I, Chapters 1-3

| publisher=Springer

| year=1998

| page=98

}}

• A set R with two binary operations + and ⋅ such that {{nowrap|(R, +)}} is an abelian group with identity 0, and {{nowrap|1=a(b + c) + a0 = ab + ac}} and {{nowrap|1=(b + c)a + 0a = ba + ca}} for all a, b, c in R.{{cite journal|last=Natarajan|first=N. S.|title=Rings with generalised distributive laws|journal=J. Indian. Math. Soc. |series=New Series|year=1964|volume=28|pages=1–6}}
• An abelian group {{nowrap|(A, +)}} equipped with a subgroup B and a multiplication {{nowrap|B × A → A}} making B a ring and A a B-module.{{cite journal|last=Patterson|first=Edward M.|title=The Jacobson radical of a pseudo-ring|journal=Math. Z.|year=1965|volume=89|issue=4|pages=348–364|doi=10.1007/bf01112167|s2cid=120796340}}

None of these definitions are equivalent, so it is best{{editorializing|date=July 2024}} to avoid the term "pseudo-ring" or to clarify which meaning is intended.