# Regular ideal

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

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

In commutative algebra a **regular ideal** refers to an ideal containing a non-zero divisor.[1] (Larsen & McCarthy 1971, p.42) This article will use "regular element ideal" to help distinguish this type of ideal.

A two-sided ideal of a ring *R* can also be called a (von Neumann) **regular ideal** if for each element *x* of there exists a *y* in 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.[2] 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 is modular if and only if 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.[3] 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.[4]

- Examples

- In the non-unital ring of even integers, (6) is regular () 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 *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.[5]

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

- .

- 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 *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.

- 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.

## References

- Non-zerodivisors in commutative rings are called
*regular elements*. - Burton, D.M. (1970) ``A first course in rings and ideals.
*Addison-Wesley. Reading, Massachusetts .* - Jacobson 1956, p.6.
- Kaplansky 1948, Lemma 1.
- Goodearl 1991, p.2.

- Goodearl, K. R. (1991),
*von Neumann regular rings*(2 ed.), Malabar, FL: Robert E. Krieger Publishing Co. Inc., pp. xviii+412, ISBN 0-89464-632-X, MR 1150975 - Jacobson, Nathan (1956),
*Structure of rings*, American Mathematical Society, Colloquium Publications, vol. 37, 190 Hope Street, Prov., R. I.: American Mathematical Society, pp. vii+263, MR 0081264 - Kaplansky, Irving (1948), "Dual rings",
*Ann. of Math.*, 2,**49**(3): 689–701, doi:10.2307/1969052, ISSN 0003-486X, JSTOR 1969052, MR 0025452 - Irving Kaplansky (1969).
*Fields and Rings*. The University of Chicago Press.

- Larsen, Max. D.; McCarthy, Paul J. (1971). "Multiplicative theory of ideals".
*Pure and Applied Mathematics*. New York: Academic Press.**43**: xiv, 298. MR 0414528. - Zhevlakov, K.A. (2001) [1994], "Modular ideal", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4