Since a maximal ideal in A is closed, is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ ( A ) of all nonzero homomorphisms from A to C. The set Δ ( A ) is called the " structure space " or " character space " of A, and its members " characters.

* In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i. e. M is contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime.

# In all unital rings, maximal ideals are prime.

Spec ( R ) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in R are precisely the closed points in this topology.

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.

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 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 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.

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.

The advantage of choosing a primitive polynomial as the generator for a CRC code is that the resulting code has maximal total block length.

If r is the degree of the primitive generator polynomial, then the maximal total block length is, and the associated code is able to detect any single-bit or double-bit errors.

If we use the generator polynomial, where is a primitive polynomial of degree, then the maximal total block length is, and the code is able to detect single, double, and triple errors.

A polynomial that admits other factorizations may be chosen then so as to balance the maximal total blocklength with a desired error detection power.

In the 1920s, Emmy Noether had first suggested a way to clarify the concept: start with the coordinate ring of the variety ( the ring of all polynomial functions defined on the variety ); the maximal ideals of this ring will correspond to ordinary points of the variety ( under suitable conditions ), and the non-maximal prime ideals will correspond to the various generic points, one for each subvariety.

There is an analogous result, also referred to as the Weierstrass preparation theorem, for power series rings over the ring of integers in a p-adic field ; namely, a power series f ( z ) can always be uniquely factored as π < sup > n </ sup >· u ( z )· p ( z ), where u ( z ) is a unit in the ring of power series, p ( z ) is a distinguished polynomial ( monic, with the coefficients of the non-leading term each in the maximal ideal ), and π is a fixed uniformizer.

NTRU is actually a parameterised family of cryptosystems ; each system is specified by three integer parameters ( N, p, q ) which represent the maximal degree for all polynomials in the truncated ring R, a small modulus and a large modulus, respectively, where it is assumed that N is prime, q is always larger than p, and p and q are coprime ; and four sets of polynomials and ( a polynomial part of the private key, a polynomial for generation of the public key, the message and a blinding value, respectively ), all of degree at most.

The problem of finding a maximal independent set can be solved in polynomial time by a trivial greedy algorithm.

A necessary and sufficient condition for the sequence generated by a LFSR to be maximal length is that its corresponding polynomial be primitive.

show that for non-negative matrices there may exist eigenvalues of the same absolute value as the maximal one (( 1 ) and (− 1 ) – eigenvalues of the first matrix ); moreover the maximal eigenvalue may not be a simple root of the characteristic polynomial, can be zero and the corresponding eigenvector ( 1, 0 ) is not strictly positive ( second example ).

The subtraction of only multiples of 2 from the maximal number of positive roots occurs because the polynomial may have complex roots, which always come in pairs since the rule applies to polynomials whose coefficients are real.

Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.

** Every unital ring other than the trivial ring contains a maximal ideal.

Let A be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A.

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 ).

The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.

The mean and maximal doses used for olanzapine were considerably higher than standard practice, and this has been postulated as a biasing factor that may explain olanzapine's superior efficacy over the other atypical antipsychotics studied, where doses were more in line with clinically relevant practices.

The Hausdorff maximal principle is one of many statements equivalent to the axiom of choice over Zermelo – Fraenkel set theory.

The maximal number of teeth is 38 with dentition formula:, but over half of the teeth are rudimentary and occur with less than 50 % frequency, such that a typical dentition includes only 18 teeth

We will go over a typical application of Zorn's lemma: the proof that every nontrivial ring R with unity contains a maximal ideal.

To see the connection with the classical picture, note that for any set S of polynomials ( over an algebraically closed field ), it follows from Hilbert's Nullstellensatz that the points of V ( S ) ( in the old sense ) are exactly the tuples ( a < sub > 1 </ sub >, ..., a < sub > n </ sub >) such that ( x < sub > 1 </ sub >-a < sub > 1 </ sub >, ..., x < sub > n </ sub >-a < sub > n </ sub >) contains S ; moreover, these are maximal ideals and by the " weak " Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form.

In fact, they agree in the sense that when A is the ring of polynomials over some algebraically closed field k, the maximal ideals of A are ( as discussed in the previous paragraph ) identified with n-tuples of elements of k, their residue fields are just k, and the " evaluation " maps are actually evaluation of polynomials at the corresponding n-tuples.

In modern language there is a maximal abelian extension A of K, which will be of infinite degree over K ; and associated to A a Galois group G which will be a pro-finite group, so a compact topological group, and also abelian.

For example, the abelianized absolute Galois group G of is ( naturally isomorphic to ) an infinite product of the group of units of the p-adic integers taken over all prime numbers p, and the corresponding maximal abelian extension of the rationals is the field generated by all roots of unity.

In an " open " or " flat " universe that continues expanding indefinitely, a heat death is also expected to occur, with the universe cooling to approach absolute zero temperature and approaching a state of maximal entropy over a very long time period.

There is dispute over whether or not an expanding universe can approach maximal entropy ; it has been proposed that in an expanding universe, the value of maximum entropy increases faster than the universe gains entropy, causing the universe to move progressively further away from heat death.

If F is any field, the separable closure F < sup > sep </ sup > of F is the field of all elements in an algebraic closure of F that are separable over F. This is the maximal Galois extension of F. By definition, F is perfect if and only if its separable and algebraic closures coincide ( in particular, the notion of a separable closure is only interesting for imperfect fields ).

If is a linear Lie algebra ( a Lie subalgebra of the Lie algebra of endomorphisms of a finite-dimensional vector space V ) over an algebraically closed field, then any Cartan subalgebra of is the centralizer of a maximal toral Lie subalgebra of ; that is, a subalgebra consisting entirely of elements which are diagonalizable as endomorphisms of V which is maximal in the sense that it is not properly included in any other such subalgebra.

This corresponds algebraically to the universal perfect central extension ( called " covering group ", by analogy ) as the maximal element, and a group mod its center as minimal element.

However, such Eisenstein equations are essentially the only ones not to have a solution, because, if K is algebraically closed of characteristic p > 0, then the field of Puiseux series over K is the perfect closure of the maximal tamely ramified extension of K.

Every character is automatically continuous from A to C, since the kernel of a character is a maximal ideal, which is closed.

( DD4 ) is an integrally closed, Noetherian domain with Krull dimension one ( i. e., every nonzero prime ideal is maximal ).

Thus, V ( S ) is " the same as " the maximal ideals containing S. Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals ; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.

* Spec &# 8484 ;, the spectrum of the integers has a closed point for every prime number p corresponding to the maximal ideal ( p ) ⊂ &# 8484 ;, and one non-closed generic point ( i. e., whose closure is the whole space ) corresponding to the zero ideal ( 0 ).

The closed points correspond to maximal ideals of A.

Among all simple closed surfaces with given surface area, the sphere encloses a region of maximal volume.

In a " closed " universe that undergoes recollapse, a heat death is expected to occur, with the universe approaching arbitrarily high temperature and maximal entropy as the end of the collapse approaches.

In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.

In that case a maximal compact subgroup K is a compact Lie group ( since a closed subgroup of a Lie group is a Lie group ), for which the theory is easier.

It follows that the systolic category of an essential n-manifold is precisely n. In fact, for closed n-manifolds, the maximal value of both the LS category and the systolic category is attained simultaneously.