Maximal ideal

Maximal ideal

In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal (with respect to set inclusion) amongst all proper ideals.[1][2] In other words, I is a maximal ideal of a ring R if I is an ideal of R, IR, and whenever J is another ideal containing I as a subset, then either J = I or J = R. So there are no ideals "in between" I and R.

Maximal ideals are important because the quotient rings of maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields.

In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal in the poset of right ideals, and similarly, a maximal left ideal is defined. Since a one sided maximal ideal A is not necessarily twosided, the quotient R/A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal twosided ideal of the ring, and is in fact the Jacobson radical J(R).

It is possible for a ring to have a unique maximal ideal and yet lack unique maximal one sided ideals: for example, in the ring of 2 by 2 square matrices over a field, the zero ideal is a maximal ideal, but there are many maximal right ideals.

Contents

Definition

Given a ring R and a proper ideal I of R (that is IR), I is called a maximal ideal of R if there exists no other proper ideal J of R so that IJ.

Equivalently, I is a maximal ideal of R if IR and for all ideals J with IJ, either J = I or J = R.

Examples

Properties

  • If R is a unital commutative ring with an ideal m, then k = R/m is a field if and only if m is a maximal ideal. In that case, R/m is known as the residue field. This fact can fail in non-unital rings. For example, 4\mathbb{Z} is a maximal ideal in 2\mathbb{Z} , but 2\mathbb{Z}/4\mathbb{Z} is not a field.
  • If L is a maximal left ideal, then R/L is a simple left R module. Conversely in rings with unity, any simple left R module arises this way. Incidentally this shows that a collection of representatives of simple left R modules is actually a set since it can be put into correspondence with part of the set of maximal left ideals of R.
  • Krull's theorem (1929): Every ring with a multiplicative identity has a maximal ideal. The result is also true if "ideal" is replaced with "right ideal" or "left ideal". More generally, it is true that every nonzero finitely generated module has a maximal submodule. Suppose I is an ideal which is not R (respectively, A is a right ideal which is not R). Then R/I is a ring with unity, (respectively, R/A is a finitely generated module), and so the above theorems can be applied to the quotient to conclude that there is a maximal ideal (respectively maximal right ideal) of R containing I (respectively, A).
  • Krull's theorem can fail for rings without unity. A radical ring, i.e. a ring in which the Jacobson radical is the entire ring, has no simple modules and hence has no maximal right or left ideals.
  • In a commutative ring with unity, every maximal ideal is a prime ideal. The converse is not always true: for example, in any nonfield integral domain the zero ideal is a prime ideal which is not maximal. Commutative rings in which prime ideals are maximal are known as zero-dimensional rings, where the dimension used is the Krull dimension.

References

  1. ^ Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. 
  2. ^ Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X. 

Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • maximal ideal — Math. an ideal in a ring that is not included in any other ideal except the ring itself. [1960 65] * * * …   Universalium

  • maximal ideal — Math. an ideal in a ring that is not included in any other ideal except the ring itself. [1960 65] …   Useful english dictionary

  • Ideal (order theory) — In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different… …   Wikipedia

  • Ideal (ring theory) — In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like even number or multiple of 3 . For instance, in… …   Wikipedia

  • Ideal maximal — Idéal maximal Richard Dedekind 1831 1916 formalisateur du concept d idéal Un idéal maximal est un concept associé à la théorie des anneaux en mathématiques et plus précisément en algèbre. Un idéal d un anneau est dit maximal si, et seulement si,… …   Wikipédia en Français

  • Idéal Maximal — Richard Dedekind 1831 1916 formalisateur du concept d idéal Un idéal maximal est un concept associé à la théorie des anneaux en mathématiques et plus précisément en algèbre. Un idéal d un anneau est dit maximal si, et seulement si, il n es …   Wikipédia en Français

  • Ideal premier — Idéal premier Richard Dedekind 1831 1916 formalisateur du concept d idéal Un idéal premier est un concept associé à la théorie des anneaux en mathématiques et plus précisément en algèbre. Un idéal d un anneau commutatif unitaire est dit premier… …   Wikipédia en Français

  • Idéal Premier — Richard Dedekind 1831 1916 formalisateur du concept d idéal Un idéal premier est un concept associé à la théorie des anneaux en mathématiques et plus précisément en algèbre. Un idéal d un anneau commutatif unitaire est dit premier si, et s …   Wikipédia en Français

  • Idéal de l'anneau des entiers d'un corps quadratique — En mathématiques et plus précisément en théorie algébrique des nombres, l anneau des entiers d un corps quadratique ressemble à certains égards à celui des entiers relatifs. Certains d entre eux sont euclidiens comme celui des entiers de Gauss d… …   Wikipédia en Français

  • Idéal maximal — Richard Dedekind 1831 1916 formalisateur du concept d idéal Un idéal maximal est un concept associé à la théorie des anneaux en mathématiques et plus précisément en algèbre. Un idéal d un anneau commutatif est dit maximal si, et seulement si, il… …   Wikipédia en Français

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”