Regular ideal

In mathematics, especially ring theory, a regular ideal can refer to multiple concepts.
In operator theory, a right ideal formula_1 in a (possibly) non-unital ring "A" is said to be regular (or modular) if there exists an element "e" in "A" such that formula_2 for every formula_3.
In commutative algebra a regular ideal refers to an ideal containing a non-zero divisor. This article will use "regular element ideal" to help distinguish this type of ideal.
A two-sided ideal formula_1 of a ring "R" can also be called a (von Neumann) regular ideal if for each element "x" of formula_1 there exists a "y" in formula_1 such that "xyx"="x".
Finally, regular ideal has been used to refer to an ideal "J" of a ring "R" such that the quotient ring "R"/"J" is von Neumann regular ring. This article will use "quotient von Neumann regular" to refer to this type of regular ideal.
Since the adjective "regular" has been overloaded, this article adopts the alternative adjectives "modular", "regular element", "von Neumann regular", and "quotient von Neumann regular" to distinguish between concepts.
Properties and examples.
Modular ideals.
The notion of modular ideals permits the generalization of various characterizations of ideals in a unital ring to non-unital settings.
A two-sided ideal formula_1 is modular if and only if formula_8 is unital. In a unital ring, every ideal is modular since choosing "e"=1 works for any right ideal. So, the notion is more interesting for non-unital rings such as Banach algebras. From the definition it is easy to see that an ideal containing a modular ideal is itself modular.
Somewhat surprisingly, it is possible to prove that even in rings without identity, a modular right ideal is contained in a maximal right ideal. 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.
Regular element ideals.
Every ring with unity has at least one regular element ideal: the trivial ideal "R" itself. Regular element ideals of commutative rings are essential ideals. In a semiprime right Goldie ring, the converse holds: essential ideals are all regular element ideals.
Since the product of two "regular elements" (=non-zerodivisors) of a commutative ring "R" is again a regular element, it is apparent that the product of two regular element ideals is again a regular element ideal. Clearly any ideal containing a regular element ideal is again a regular element ideal.
Von Neumann regular ideals.
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". Proof: The "only if" direction is a tautology. For the "if" direction, we have "a"="ara"="arara". Since "a" is in "J", so is "rar", and so by setting "y"="rar" we have the conclusion.
As a consequence of this lemma, it is apparent that every ideal of a von Neumann regular ring is a von Neumann regular ideal. Another consequence is that if "J" and "K" are two ideals of "R" such that "J"⊆"K" and "K" is a von Neumann regular ideal, then "J" is also a von Neumann regular ideal.
If "J" and "K" are two ideals of "R", then "K" is von Neumann regular if and only if both "J" is a von Neumann regular ideal and "K"/"J" is a von Neumann regular ring.
Every ring has at least one von Neumann regular ideal, namely {0}. Furthermore, every ring has a maximal von Neumann regular ideal containing all other von Neumann regular ideals, and this ideal is given by
Quotient von Neumann regular ideals.
If "J" and "K" are quotient von Neumann regular ideals, then so is "J"∩"K".
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"). In particular if "A" is "any" ideal in "R" the ideal "A"+"J" is quotient von Neumann regular if "J" is.