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In [[mathematics]], and more specifically in [[abstract algebra]], a '''pseudo-ring''' is one of the following variants of a [[ring (mathematics)|ring]]:

* A [[rng (algebra)|rng]], i.e., a structure satisfying all the axioms of a ring except for the existence of a [[multiplicative identity]].<ref>{{Cite book
 | last1=Bourbaki
 | first1=N.
 | author1-link=Nicolas Bourbaki
 | title=Algebra I, Chapters 1-3
 | publisher=Springer
 | year=1998
 | page=98
}}</ref>
* A set ''R'' with two [[binary operation]]s + and · such that (''R'',+) is an [[abelian group]] with [[additive identity|identity]] 0, and <math>a(b+c) + a0 =ab +ac</math> and <math>(b+c)a +0a=ba+ca</math> for all ''a'', ''b'', ''c'' in ''R''.<ref name=Natarajan>{{cite journal|last=Natarajan|first=N. S.|title=Rings with generalised distributive laws|journal=J. Indian. Math. Soc. (N. S.)|year=1964|volume=28|pages=1-6}}</ref>
* An abelian group (''A'',+) equipped with a [[subgroup]] ''B'' and a multiplication ''B'' × ''A'' → ''A'' making ''B'' a ring and ''A'' a ''B''-[[module (mathematics)|module]].<ref name=Patterson>{{cite journal|last=Patterson|first=Edward M.|title=The Jacobson radical of a pseudo-ring|journal=Math. Z.|year=1965|volume=89|pages=348-364}}</ref> 

No two of these definitions are equivalent, so it is best to avoid the term "pseudo-ring" or to clarify which meaning is intended.

== References ==
{{reflist}}

{{DEFAULTSORT:Pseudo-ring}}
[[Category:Ring theory]]
[[Category:Algebraic structures]]
[[Category:Algebras]]