In mathematics, especially ring theory, a regular ideal can refer to multiple concepts.
in a (possibly) non-unital ring A is said to be regular (or modular) if there exists an element e in A such that
of a ring R can also be called a (von Neumann) regular ideal if for each element x of
So, the notion is more interesting for non-unital rings such as Banach algebras.
[7] However, it is possible for a ring without identity to lack modular right ideals entirely.
The intersection of all maximal right ideals which are modular is the Jacobson radical.
From the definition, it is clear that R is a von Neumann regular ring if and only if R is a von Neumann regular ideal.
The following statement is a relevant lemma for von Neumann regular ideals: Lemma: For a ring R and proper ideal J containing an element a, there exists and element y in J such that a=aya if and only if there exists an element r in R such that a=ara.
As a consequence of this lemma, it is apparent that every ideal of a von Neumann regular ring is a von Neumann regular ideal.
[10] Every ring has at least one von Neumann regular ideal, namely {0}.
If J⊆K are proper ideals of R and J is quotient von Neumann regular, then so is K. This is because quotients of R/J are all von Neumann regular rings, and an isomorphism theorem for rings establishing that R/K≅(R/J)/(J/K).