# Regular ideal

In mathematics, especially ring theory, a regular ideal can refer to multiple concepts.

In operator theory, a right ideal ${\mathfrak {i}}$ in a (possibly) non-unital ring A is said to be regular (or modular) if there exists an element e in A such that $ex-x\in {\mathfrak {i}}$ for every $x\in A$ . (Jacobson 1956)

In commutative algebra a regular ideal refers to an ideal containing a non-zero divisor. (Larsen & McCarthy 1971, p.42) This article will use "regular element ideal" to help distinguish this type of ideal.

A two-sided ideal ${\mathfrak {i}}$ of a ring R can also be called a (von Neumann) regular ideal if for each element x of ${\mathfrak {i}}$ there exists a y in ${\mathfrak {i}}$ such that xyx=x. (Goodearl 1991, p.2) (Kaplansky 1969, p.112)

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.

## 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 ${\mathfrak {i}}$  is modular if and only if $A/{\mathfrak {i}}$  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.

Examples
• In the non-unital ring of even integers, (6) is regular ($e=4$ ) while (4) is not.
• Let M be a simple right A-module. If x is a nonzero element in M, then the annihilator of x is a regular maximal right ideal in A.
• If A is a ring without maximal right ideals, then A cannot have even a single modular right ideal.

### 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. (Lam 1999, p.342)

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.

Examples
• In an integral domain, every nonzero element is a regular element, and so every nonzero ideal is a regular element ideal.
• The nilradical of a commutative ring is composed entirely of nilpotent elements, and therefore no element can be regular. This gives an example of an ideal which is not a regular element ideal.
• In an Artinian ring, each element is either invertible or a zero divisor. Because of this, such a ring only has one regular element ideal: just R.

### 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 JK 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

$M=\{x\in R\mid RxR{\text{ is a von Neumann regular ideal }}\}$ .
Examples
• As noted above, every ideal of a von Neumann regular ring is a von Neumann regular ideal.
• It is well known that a local ring which is also a von Neumann regular ring is a division ring. Let R Be a local ring which is not a division ring, and denote the unique maximal right ideal by J. Then R cannot be von Neumann regular, but R/J, being a division ring, is a von Neumann regular ring. Consequently, J cannot be a von Neumann regular ideal, even though it is maximal.
• A simple domain which is not a division ring has the minimum possible number of von Neumann regular ideals: only the {0} ideal.

### Quotient von Neumann regular ideals

If J and K are quotient von Neumann regular ideals, then so is JK.

If JK 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.

Examples
• Every proper ideal of a von Neumann regular ring is quotient von Neumann regular.
• Any maximal ideal in a commutative ring is a quotient von Neumann regular ideal since R/M is a field. This is not true in general because for noncommutative rings R/M may only be a simple ring, and may not be von Neumann regular.
• Let R be a local ring which is not a division ring, and with maximal right ideal M . Then M is a quotient von Neumann regular ideal, since R/M is a division ring, but R is not a von Neumann regular ring.
• More generally in any semilocal ring the Jacobson radical J is quotient von Neumann regular, since R/J is a semisimple ring, hence a von Neumann regular ring.