Regular ideal

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).