In
mathematics, more specifically
ring theory, a left, right or two-sided
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 considered ...
of a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
is said to be a nil ideal if each of its elements is
nilpotent
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 ...
.
[, p. 194]
The
nilradical of a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil. Unfortunately the set of nil elements does not always form an ideal for
noncommutative ring
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...
s. Nil ideals are still associated with interesting open questions, especially the unsolved
Köthe conjecture.
Commutative rings
In commutative rings, the nil ideals are better understood than in noncommutative rings, primarily because in commutative rings, products involving
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 sums of nilpotent elements are both nilpotent. This is because if ''a'' and ''b'' are nilpotent elements of ''R'' with ''a''
n=0 and ''b''
m=0, and r is any element of R, then (''a''·''r'')
n = ''a''
n·''r''
n = 0, and by the binomial theorem, (''a''+''b'')
m+n=0. Therefore, the set of all nilpotent elements forms an ideal known as the nilradical of a ring. Because the nilradical contains every nilpotent element, an ideal of a commutative ring is nil if and only if it is a subset of the nilradical, and so the nilradical is maximal among nil ideals. Furthermore, for any nilpotent element ''a'' of a commutative ring ''R'', the ideal ''aR'' is nil. For a noncommutative ring however, it is not in general true that the set of nilpotent elements forms an ideal, or that ''a''·''R'' is a nil (one-sided) ideal, even if ''a'' is nilpotent.
Noncommutative rings
The theory of nil ideals is of major importance in noncommutative ring theory. In particular, through the understanding of
nil rings—rings whose every element is nilpotent—one may obtain a much better understanding of more general rings.
In the case of commutative rings, there is always a maximal nil ideal: the nilradical of the ring. The existence of such a maximal nil ideal in the case of noncommutative rings is guaranteed by the fact that the sum of nil ideals is again nil. However, the truth of the assertion that the sum of two left nil ideals is again a left nil ideal remains elusive; it is an open problem known as the
Köthe conjecture.
[, p. 21] The Köthe conjecture was first posed in 1930 and yet remains unresolved as of 2022.
Relation to nilpotent ideals
The notion of a nil ideal has a deep connection with that of a
nilpotent ideal, and in some classes of rings, the two notions coincide. If an ideal is nilpotent, it is of course nil. There are two main barriers for nil ideals to be nilpotent:
# There need not be an upper bound on the exponent required to annihilate elements. Arbitrarily high exponents may be required.
# The product of ''n'' nilpotent elements may be nonzero for arbitrarily high ''n''.
Clearly both of these barriers must be avoided for a nil ideal to qualify as nilpotent.
In a
right artinian ring, any nil ideal is nilpotent. This is proven by observing that any nil ideal is contained in the
Jacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal (due to the artinian hypothesis), the result follows. In fact, this has been generalized to
right noetherian rings; the result is known as
Levitzky's theorem. A particularly simple proof due to Utumi can be found in .
See also
*
Köthe conjecture
*
Nilpotent ideal
*
Nilradical
*
Jacobson radical
Notes
References
*
*
*{{citation
, last=Smoktunowicz
, first=Agata , authorlink = Agata Smoktunowicz
, contribution=Some results in noncommutative ring theory
, title=International Congress of Mathematicians, Vol. II
, year=2006
, isbn=978-3-03719-022-7
, publisher=
European Mathematical Society
, place=Zürich
, mr=2275597
, pages=259–269
, contribution-url=http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_12.pdf
, accessdate=2009-08-19
Ideals (ring theory)