* The Artin – Wedderburn theorem determines the structure of semisimple rings.

* The Jacobson density theorem determines the structure of primitive rings.

* Goldie's theorem determines the structure of semiprime Goldie rings.

One such case is Fermat's theorem on sums of two squares, which determines when an integer may be expressed in the form, where x, y are integers.

As the Shannon-Hartley theorem shows, it is a combination of bandwidth and signal-to-noise ratio which determines the maximum information rate of a channel.

Although Desargues ' theorem chooses different roles for these ten lines and points, the Desargues configuration itself is more symmetric: any of the ten points may be chosen to be the center of perspectivity, and that choice determines which six points will be the vertices of triangles and which line will be the axis of perspectivity.

The fundamental theorem of curves asserts that the knowledge of these invariants completely determines the curve.

The exponential of its vacuum expectation value determines the coupling constant g, as for compact worldsheets by the Gauss-Bonnet theorem and the Euler characteristic, where g is the genus that counts the number of handles and thus the number of loops or string interactions described by a specific worldsheet.

Girard's theorem, named after the 16th century French mathematician Albert Girard ( earlier discovered but not published by the English mathematician Thomas Harriot ), states that this surplus determines the surface area of any spherical triangle:

As Paul Erdős observed, the Sylvester – Gallai theorem immediately implies that any set of n points that are not collinear determines at least n different lines.

* The de Bruijn – Erdős theorem, a consequence of this theorem, states that a set of n noncollinear points determines n lines.

In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V. It says, roughly speaking, that the space spanned by such curves ( up to linear equivalence ) has a one-dimensional subspace on which it is positive definite ( not uniquely determined ), and decomposes as a direct sum of some such one-dimensional subspace, and a complementary subspace on which it is negative definite.

The sum of the angles of the triangle determines the type of the geometry by the Gauss – Bonnet theorem: it is Euclidean if the angle sum is exactly π, spherical if it exceeds π and hyperbolic if it is strictly smaller than π.

Employing the Schrödinger equation as its starting point, the Runge-Gross theorem shows that at any time, the density uniquely determines the external potential.

For a given interaction potential, the RG theorem shows that the external potential uniquely determines the density.

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.

The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups ( and their action on other mathematical objects ) can sometimes be reduced to questions about finite simple groups.

John Milnor discovered that some spheres have more than one smooth structure — see exotic sphere and Donaldson's theorem.

In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.

Well-known applications include automatic theorem proving and modeling the elaboration of linguistic structure.

The proof of this statement uses the linearity of classical probability, and has exactly the same structure as the proof of the quantum no-cloning theorem.

The key result is the structure theorem: If R is a principal ideal domain, and M is a finitely

The oldest result relating algebraic structure to solvability of the word problem is Kuznetsov's theorem:

While the ideal structure of becomes considerably more complex as n increases, the rings in question still remain Noetherian, and any theorem about that can be proven using only the fact that is Noetherian, can be proven for.

From the mid 50's he moved into singularity theory, of which catastrophe theory is just one aspect, and in a series of deep ( and at the time obscure ) papers between 1960 and 1969 developed the theory of stratified sets and stratified maps, proving a basic stratified isotopy theorem describing the local conical structure of Whitney stratified sets, now known as the Thom-Mather isotopy theorem.

In view of the well known and exceedingly useful structure theorem for finitely generated modules over a principal ideal domain ( PID ), it is natural to ask for a corresponding theory for finitely generated modules over a Dedekind domain.

Dedekind groups are named after Richard Dedekind, who investigated them in, proving a form of the above structure theorem ( for finite groups ).

Moreover Mazur's torsion theorem restricts the structure of the torsion subgroup.

The Cartan – Dieudonné theorem describes the structure of the orthogonal group for a non-singular form.

The no-hair theorem is formulated in the classical spacetime of Einstein's general relativity, assumed to be infinitely divisible with no limiting short-range structure or short-range correlations.

The concept of dual cell structure, which Henri Poincaré used in his proof of his Poincaré duality theorem, contained the germ of the idea of cohomology, but this was not seen until later.

In a deeper way, this theorem relates the topology of the domain of integration to the structure of the differential forms themselves ; the precise connection is known as De Rham's theorem.

Normal subgroups of prime power index are kernels of surjective maps to p-groups and have interesting structure, as described at Focal subgroup theorem: Subgroups and elaborated at focal subgroup theorem.

Even though the topological structure of Fréchet spaces is more complicated than that of Banach spaces due to the lack of a norm, many important results in functional analysis, like the Hahn – Banach theorem, the open mapping theorem, and the Banach – Steinhaus theorem, still hold.

However, by Wedderburn's little theorem all finite division rings are commutative and therefore finite fields.

Wedderburn's little theorem: All finite division rings are commutative and therefore finite fields.

In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.

The theorem follows because * is ( commutative and ) associative, and 1 * μ = i, where i is the identity function for the Dirichlet convolution, taking values i ( 1 )= 1, i ( n )= 0 for all n > 1.

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.

The transfer principle states that true first order statements about R are also valid in * R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals ; since R is a real closed field, so is * R. Since for all integers n, one also has for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś ' theorem of 1955.

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 more abstract language, the spectral theorem is a statement about commutative C *- algebras.

In linear algebra, the Cayley – Hamilton theorem ( named after the mathematicians Arthur Cayley and William Hamilton ) states that every square matrix over a commutative ring ( such as the real or complex field ) satisfies its own characteristic equation.

We shall therefore now consider only arguments that prove the theorem directly for any matrix using algebraic manipulations only ; these also have the benefit of working for matrices with entries in any commutative ring.

The proof of the general case requires some commutative algebra, namely the fact, that the Krull dimension of k is one — see Krull's principal ideal theorem ).

Hence, by the structure theorem for finitely generated abelian groups, it is isomorphic to a product of a free abelian group Z < sup > r </ sup > and a finite commutative group for some non-negative integer r called the rank of the abelian variety.

For example, Hilbert's Nullstellensatz is a theorem which is fundamental for algebraic geometry, and is stated and proved in term of commutative algebra.

Similarly, Fermat's last theorem is stated in term of elementary arithmetic, which is a part of commutative algebra, but its proof involves deep results of both algebraic number theory and algebraic geometry.

There are also generalizations of the Cone and of the Proj of a commutative graded ring, mimicking a Serre's theorem on Proj.

Namely the category of quasicoherent sheaves of O-modules on a Proj of a commutative graded algebra is equivalent to the category of graded modules over the ring localized on Serre's subcategory of graded modules of finite length ; there is also analogous theorem for coherent sheaves when the algebra is Noetherian.

Here Bishop worked on uniform algebras ( commutative Banach algebras with unit whose norms are the spectral norms ) proving results such as antisymmetric decomposition of a uniform algebra, the Bishop-DeLeeuw theorem, and the proof of existence of Jensen measures.

By the Gelfand theorem, a commutative C *- algebra is isomorphic to the C *- algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C *- algebra up to homeomorphism.

Satisfying Pappus's theorem universally is equivalent to having the underlying coordinate system be commutative.

The Weierstrass preparation theorem would now be classed as commutative algebra ; it did justify the local picture, ramification, that addresses the generalisation of the branch points of Riemann surface theory.

In the mathematical fields of topology and K-theory, the Serre – Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: " projective modules over commutative rings are like vector bundles on compact spaces ".