* The spectrum of any commutative ring with the Zariski topology ( that is, the set of all prime ideals ) is compact, but never Hausdorff ( except in trivial cases ).

Should a president be of one side of the political spectrum and the opposition be in control of the legislature, the president is usually obliged to select someone from the opposition to become prime minister, a process known as Cohabitation.

In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec ( R ), is the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.

It follows readily from the definition of the spectrum of a ring, the space of prime ideals of equipped with the Zariski topology, that the Krull dimension of is precisely equal to the irreducible dimension of its spectrum.

* In the category of schemes, Spec ( Z ) the prime spectrum of the ring of integers is a terminal object.

The empty scheme ( equal to the prime spectrum of the trivial ring ) is an initial object.

One prime example is the invention of the linear phase equalizer, which has inherent phase shift that is homogeneous across the frequency spectrum.

Thames is often quoted as a prime example of a good commercial public-service broadcaster with shows covering all aspects of the spectrum and the largest producer in the network.

Modern algebraic geometry takes the spectrum of a ring ( the set of proper prime ideals ) as its starting point.

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 ).

* Spec k, the spectrum of the polynomial ring over a field k, which is also denoted, the affine line: the polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k. If k is algebraically closed, for example the field of complex numbers, a non-constant polynomial is irreducible if and only if it is linear, of the form t − a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal.

Note, however, that the spectrum and projective spectrum are still T < sub > 0 </ sub > spaces: given two points P, Q, which are prime ideals of A, at least one of them, say P, does not contain the other.

Let P be a finitely generated projective module over a commutative ring R and X be the spectrum of R. The rank of P at a prime ideal in X is the rank of the free-module.

He defined the spectrum of a commutative ring as the space of prime ideals with Zariski topology, but augments it with a sheaf of rings: to every Zariski-open set he assigns a commutative ring, thought of as the ring of " polynomial functions " defined on that set.

The functor associates to every commutative ring its spectrum, the scheme defined by the prime ideals of the ring.

* Affine scheme, the spectrum of prime ideals of a commutative ring

Andimuthu Raja, the prime accused in the 2G spectrum scam, also revealed on tapped phones that Chidambaram received a lot of money for 2G spectrum allocation.

* In commutative algebra, a commutative ring R is irreducible if its prime spectrum, that is, the topological space Spec R, is an irreducible topological space.

For every prime number p ( which is suppressed in the notation ), it consists of theories K ( n ) for each nonnegative integer n, each a ring spectrum in the sense of homotopy theory.

Another way, perhaps more similar to the original, to interpret the modern definition is to realize that the elements of A can actually be thought of as functions on the prime ideals of A ; namely, as functions on Spec A.

Therefore, the induced map on spectra Spec ( B < sub > q < sub > 2 </ sub ></ sub >) → Spec ( A < sub > p < sub > 2 </ sub ></ sub >) is surjective and there exists a prime ideal of B < sub > q < sub > 2 </ sub ></ sub > that contracts to the prime ideal p < sub > 1 </ sub > A < sub > p < sub > 2 </ sub ></ sub > of A < sub > p < sub > 2 </ sub ></ sub >.

There is the following description of irreducible affine varieties or schemes X = Spec A: X is irreducible iff the coordinate ring A of X has one minimal prime ideal.

In particular, if A has no zero divisors, Spec A is irreducible, because then the zero-ideal is the minimal prime ideal.

Although not officially announced, the original backup crew consisted of Fred Haise ( CDR ), William R. Pogue ( CMP ) and Gerald Carr ( LMP ) who were targeted for the prime crew assignment on Apollo 19.

The space Q < sub > p </ sub > of p-adic numbers is complete for any prime number p. This space completes Q with the p-adic metric in the same way that R completes Q with the usual metric.

Alan Turing searched for them on the Manchester Mark 1 in 1949, but the first successful identification of a Mersenne prime, M < sub > 521 </ sub >, by this means was achieved at 10: 00 P. M. on January 30, 1952 using the U. S. National Bureau of Standards Western Automatic Computer ( SWAC ) at the Institute for Numerical Analysis at the University of California, Los Angeles, under the direction of Lehmer, with a computer search program written and run by Prof. R. M.

Another example is the construction of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime number as the coefficient ring R ( see modular arithmetic ).

Prime numbers give rise to two more general concepts that apply to elements of any commutative ring R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements.

An element p of R is called prime element if it is neither zero nor a unit ( i. e., does not have a multiplicative inverse ) and satisfies the following requirement: given x and y in R such that p divides the product xy, then p divides x or y.

In any ring R, any prime element is irreducible.

It is equivalent to saying that the prime decomposition of the manifold has no acyclic components, and turns out to be equivalent to the condition that all geometric pieces of the manifold have geometries based on the two Thurston geometries S < sup > 2 </ sup >× R and S < sup > 3 </ sup >.

An ideal P of a commutative ring R is prime if it has the following two properties:

* If R denotes the ring CY of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y < sup > 2 </ sup > − X < sup > 3 </ sup > − X − 1 is a prime ideal ( see elliptic curve ).

* 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.

* An ideal I in the ring R is prime if and only if the factor ring R / I is an integral domain.

The ideal A is a prime ideal in R if all but one of the A < sub > i </ sub > are equal to R < sub > i </ sub > and the remaining A < sub > i </ sub > is a prime ideal in R < sub > i </ sub >.

For example, the direct sum of the R < sub > i </ sub > form an ideal not contained in any such A, but the axiom of choice gives that it is contained in some maximal ideal which is a fortiori prime.

S. W. R. D. Bandaranaike was elected prime minister in 1956.

Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry.

The prime ideals of the ring of integers are the ideals ( 0 ), ( 2 ), ( 3 ), ( 5 ), ( 7 ), ( 11 ), … The fundamental theorem of arithmetic generalizes to the Lasker – Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.

* In a commutative ring with unity, every maximal ideal is a prime ideal.

In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull ( 1899 – 1971 ), is the supremum of the number of strict inclusions in a chain of prime ideals.

For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated.

Krull's principal ideal theorem states that every principal ideal in a commutative Noetherian ring has height one ; that is, every principal ideal is contained in a prime ideal minimal amongst nonzero prime ideals.

In mathematics, a unique factorization domain ( UFD ) is a commutative ring in which every non-unit element, with special exceptions, can be uniquely written as a product of prime elements ( or irreducible elements ), analogous to the fundamental theorem of arithmetic for the integers.

In abstract algebra, an element of a commutative ring is said to be prime if it is not zero or a unit and whenever divides for some and in, then divides or divides.

( A non-zero non-unit element in a commutative ring is called prime if whenever for some and in, then or.

In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal.

In commutative ring theory, numbers are often replaced by ideals, and the definition of prime ideal tries to capture the essence of prime numbers.

In commutative algebraic geometry, algebraic schemes are locally prime spectra of commutative unital rings ( A. Grothendieck ), and schemes can be reconstructed from the categories of quasicoherent sheaves of modules on them ( P. Gabriel-A.

They generalize to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry.

Technically, a scheme is a topological space together with commutative rings for all of its open sets, which arises from gluing together spectra ( spaces of prime ideals ) of commutative rings along their open subsets.