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.

We can put a topology on Spec ( R ) by defining the collection of closed sets to be

For f ∈ R, define D < sub > f </ sub > to be the set of ideals of R not containing f. Then each D < sub > f </ sub > is an open subset of Spec ( R ), and is a basis for the Zariski topology.

However, Spec ( R ) is always a Kolmogorov space.

Given the space X = Spec ( R ) with the Zariski topology, the structure sheaf O < sub > X </ sub > is defined on the D < sub > f </ sub > by setting Γ ( D < sub > f </ sub >, O < sub > X </ sub >) = R < sub > f </ sub >, the localization of R at the multiplicative system

Zero divisors have a topological interpretation, at least in the case of commutative rings: a ring R is an integral domain, if and only if it is reduced and its spectrum Spec R is an irreducible topological space.

: Spec ( R / I ) → Spec ( R )

: Spec ( R / I < sup > 2 </ sup >) → Spec ( R )

More generally, when X is an affine scheme Spec ( R ), the invertible sheaves come from projective modules over R, of rank 1.

The video game Spec Ops: The Line, released on 26 June 2012, is a loose, modernized adaptation of Heart of Darkness.

Then Spec ( A < sub > S </ sub >) is homeomorphic to

Inducing oversteer is easy and very controllable, thus making the MX-5 a popular choice for amateur and stock racing, including, in the US, the Sports Car Club of America's Solo2 autocross and Spec Miata race series, and in the UK, the Ma5da racing championship.

This approach is the basis of at least five proof-of-correctness systems for computer programs, beginning with the Stanford Pascal Verifier in the late 1970s and continuing though to Microsoft's Spec # system of 2005.

Some times a DFS ( Departure From Spec ) is required to operate machinery in a degraded state.

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

Adjacent to Melching Field is Spec Martin Stadium, a 6, 000-seat football stadium.

As part of Stetson's re-entry into college football, Spec Martin Stadium is scheduled to undergo significant renovations, including a new pressbox, handicap and premium seating areas, and new locker room facilities.

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.

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.

* Spec k, the spectrum of a field k is the topological space with one element.

* 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.

The prime spectrum Spec ( R ) of a commutative ring R with the Zariski topology is a compact sober T < sub > 0 </ sub > space.

In fact, every spectral space ( i. e. a compact sober space for which the collection of compact open subsets is closed under finite intersections and forms a base for the topology ) is homeomorphic to Spec ( R ) for some commutative ring R. This is a theorem of Melvin Hochster.

The Nissan Sentra SE-R Spec V is the sport compact version of the Nissan Sentra ; it is the highest trim level of the Sentra.

So the closed subsets of Spec &# 8484 ; are precisely finite unions of closed points and the whole space.

For any field, the closed subsets of Spec k are finite unions of closed points, and the whole space.

is an s-dimensional affine space over Spec k.

in the parlance of schemes, morphisms Spec K / to a scheme X over K correspond to a choice of a rational point x ∈ X ( k ) and an element of the tangent space.

The projective space P < sup > d </ sup > over a field K is proper over a point ( that is, Spec ( K )).

* 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.

In space, Morgan was able to utilize his gunbarrels to fight Edward and defeated his GAT-333 Raider Full Spec ; however, Edward managed to escape by riding a wave to Earth, similar to how Lowe Guele did.

* MacBinary II + Spec, a proposed enhancement to MacBinary to incorporate a directory tree, never widely adopted.

There is some speculation surrounding the Spec V designation being a play on the GT-R V-Spec Skyline name, the speculation has been proven to be true thanks to the introduction of the R35 GT-R Spec V making the connection obvious fact.

The Spec V designation is due to the fact that for some unknown legal reason Nissan cannot use the V-Spec title in the US market.

Also adding to the appeal is the fact that A Spec Miata can be raced in both SM and ITA / ITS in SCCA Regional Competition.

# There is a quasi-coherent sheaf of ideals on X such that and f is an isomorphism of Z onto the global Spec of over X.

Spec scripts are usually penned by unknown professional screenwriters and amateur screenwriters.

Vehicles destined for the Middle East are referred to as " GCC Spec " vehicles, and the Crown Victoria is available in five different trim levels in Gulf Cooperation Council ( GCC ) countries:

At or adjacent to McCulloch are the Charles Bowles Track used for track meets and Roy S. " Spec " Keene Stadium where baseball plays its home games.

Cali Spec emissions control are three catalytic converters and four O < sub > 2 </ sub > sensors.

Noting that Spec K is the generic point of Spec R and discrete valuation rings are precisely the regular local one-dimensional rings, one may rephrase the criterion: given a regular curve on Y ( corresponding to the morphism s: Spec R → Y ) and given a lift of the generic point of this curve to X, f is proper if and only if there is exactly one way to complete the curve.

Spec scripts are feature film or television show scripts written on speculation, without the commission of a film studio, production company, or TV network.

# Spec Motor-Spec motors are built to the series ' specifications.

In particular, if the variety V = Spec A is smooth, cyclic cohomology of A are expressed in terms of the de Rham cohomology of V as follows:

Forthcoming are stories in Twisted Cat Tales, Dred, Space Squid and Electric Spec.

The lowest-level x86 interrupt dispatch code is written in assembly language and C. Once this code has done its job, it invokes the kernel, whose runtime and garbage collector are written in Sing # ( an extended version of Spec #, itself an extension of C #) and runs in unprotected mode.

As in the case of the Spec construction there are many ways to proceed: the most direct one, which is also highly suggestive of the construction of regular functions on a projective variety in classical algebraic geometry, is the following.