In
mathematics, especially
ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, a regular ideal can refer to multiple concepts.
In
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...
, a right
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
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
.
In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
a regular ideal refers to an ideal containing a non-
zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zer ...
. 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''.
Finally, regular ideal has been used to refer to an ideal ''J'' of a ring ''R'' such that the
quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
''R''/''J'' is
von Neumann regular ring
In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the elemen ...
.
[Burton, D.M. (1970) ''A first course in rings and ideals.'' Addison-Wesley. Reading, Massachusetts .] 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 algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
s. 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 In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition y ...
.
;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
In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers.
Because there are infinitely many prime ...
right
Goldie ring
In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring ''R'' that has finite uniform dimension (="finite rank") as a right module o ...
, 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.
;Examples
* In an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
, 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 element
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
s, 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 In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are ...
, each element is either
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
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
In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the elemen ...
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 . 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 In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
which is also a von Neumann regular ring is a
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
. 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
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
which is not a division ring has the minimum possible number of von Neumann regular ideals: only the 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
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist ...
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
In mathematics, a semi-local ring is a ring for which ''R''/J(''R'') is a semisimple ring, where J(''R'') is the Jacobson radical of ''R''.
The above definition is satisfied if ''R'' has a finite number of maximal right ideals (and finite numb ...
the
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition y ...
''J'' is quotient von Neumann regular, since ''R''/''J'' is a
semisimple ring
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itsel ...
, hence a von Neumann regular ring.
References
Bibliography
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{{DEFAULTSORT:Regular Ideal
Ring theory
Ideals (ring theory)