Ideal of a ring contained in no other ideal except the ring itself
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that 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 there are no other ideals contained between I and R.
Maximal ideals are important because the quotients of rings by 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 element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one-sided maximal ideal A is not necessarily two-sided, 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 two-sided ideal of the ring, and is in fact the Jacobson radical J(R).
It is possible for a ring to have a unique maximal two-sided 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 two-sided ideal, but there are many maximal right ideals.
^Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
^Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.
theory, a maximalideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximalideal of a ring...
and the whole ring R. Every maximalideal is in fact prime. In a principal ideal domain every nonzero prime ideal is maximal, but this is not true in general...
appropriate notions of ideals, for example, rings and prime ideals (of ring theory), or distributive lattices and maximalideals (of order theory). This...
following equivalent properties: R has a unique maximal left ideal. R has a unique maximal right ideal. 1 ≠ 0 and the sum of any two non-units in R is...
{Spec} (R)} is a compact space, but almost never Hausdorff: in fact, the maximalideals in R are precisely the closed points in this topology. By the same reasoning...
principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximalideal is nilpotent...
domain. This is because an ideal is prime if and only if the quotient of the ring by the ideal is an integral domain. Maximalideals of k[V] correspond to...
inverse in D, and they form an ideal M. This ideal is maximal among the (totally ordered) ideals of D. Since M is a maximalideal, the quotient ring D/M is...
topology such that a set of maximalideals is closed if and only if it is the set of all maximalideals that contain a given ideal. Another basic idea of Grothendieck's...
catenary if all maximal chains between two prime ideals have the same length. center The center of a valuation (or place) is the ideal of elements of positive...
Artinian ring, every maximalideal is a minimal prime ideal. In an integral domain, the only minimal prime ideal is the zero ideal. In the ring Z of integers...
discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximalideal. This means a DVR is an integral domain R that...
number field) any ideal (such as the one generated by 6) decomposes uniquely as a product of prime ideals. Any maximalideal is a prime ideal or, more briefly...
ideal is a left primitive ring. For commutative rings the primitive ideals are maximal, and so commutative primitive rings are all fields. The primitive...
-primary. An ideal whose radical is maximal, however, is primary. Every ideal Q with radical P is contained in a smallest P-primary ideal: all elements...
and only if the maximalideal of R is divisorial. An integral domain that satisfies the ascending chain conditions on divisorial ideals is called a Mori...
that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximalideals so in this case a...
{\displaystyle x\in X} the maximalideal of the local ring (stalk) at f ( x ) ∈ Y {\displaystyle f(x)\in Y} is mapped into the maximalideal of the local ring...
the powers of a proper ideal I determines the Krull (after Wolfgang Krull) or I-adic topology on R. The case of a maximalideal I = m {\displaystyle I={\mathfrak...
maximalideal, then the residue field is the quotient ring k = R/m, which is a field. Frequently, R is a local ring and m is then its unique maximal ideal...
Global Information