** Krull's intersection theorem

* Krull's theorem ( 1929 ): Every ring with a multiplicative identity has a maximal ideal.

* Krull's theorem can fail for rings without unity.

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.

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 ).

In a Noetherian ring, Krull's height theorem says that the height of an ideal generated by n elements is no greater than n.

# REDIRECT Krull's principal ideal theorem

In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull ( 1899 – 1971 ), gives a bound on the height of a principal ideal in a Noetherian ring.

This theorem can be generalized to ideals that are not principal, and the result is often called Krull's height theorem.

# REDIRECT Krull's principal ideal theorem

** Krull's principal ideal theorem

** Krull's theorem

* Krull's principal ideal theorem

* Krull's theorem

Then its coefficients generate a proper ideal I, which by Krull's theorem ( which depends on the axiom of choice ) is contained in a maximal ideal m of R. Then R / m is a field, and ( R / m ) is therefore an integral domain.

# Yoga of the Grothendieck – Riemann – Roch theorem ( K-theory, relation with intersection theory ).

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.

For instance, the ring of integers and the polynomial ring over a field are both Noetherian rings, and consequently, such theorems as the Lasker – Noether theorem, the Krull intersection theorem, and the Hilbert's basis theorem hold for them.

Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves.

Bezout's theorem was essentially stated by Isaac Newton in his proof of lemma 28 of volume 1 of his Principia, where he claims that two curves have a number of intersection points given by the product of their degrees.

The most delicate part of Bézout's theorem and its generalization to the case of k algebraic hypersurfaces in k-dimensional projective space is the procedure of assigning the proper intersection multiplicities.

Furthermore because of the Feuerbach conic theorem mentioned above, there exists a unique rectangular circumconic, centered at the common intersection point of the four nine-point circles, that passes through the four original arbitrary points as well as the orthocenters of the four triangles.

Because of the Mordell – Weil theorem, Faltings ' theorem can be reformulated as a statement about the intersection of a curve C with a finitely generated subgroup Γ of an abelian variety A. Generalizing by replacing C by an arbitrary subvariety of A and Γ by an arbitrary finite-rank subgroup of A leads to the Mordell – Lang conjecture, which has been proved.

* Brahmagupta's theorem states that for a cyclic quadrilateral that is also orthodiagonal, the perpendicular from any side through the point of intersection of the diagonals bisects the opposite side.

* Transversality theorem, a concept related to the intersection of manifolds in topology

So construct the midpoint H between O and P, and draw a circle centered at H through O and P. By Thales ' theorem, the sought point T is the intersection of this circle with the given circle k, because that is the point on k that completes a right triangle OTP.

Monge's theorem also asserts that three points lie on a line, and has a proof using the same idea of considering it in three rather than two dimensions and writing the line as an intersection of two planes.

The ten lines involved in Desargues ' theorem ( six sides of triangles, the three lines Aa, Bb, and Cc, and the axis of perspectivity ) and the ten points involved ( the six vertices, the three points of intersection on the axis of perspectivity, and the center of perspectivity ) are so arranged that each of the ten lines passes through three of the ten points, and each of the ten points lies on three of the ten lines.

The intersection of two cubics, which is points ( by Bézout's theorem ), is special in that nine points in general position are contained in a unique cubic, while if they are contained in two cubics they in fact are contained in a pencil ( 1-parameter linear system ) of cubics, whose equations are the projective linear combinations of the equations for the two cubics.

One needs a definition of intersection number in order to state results like Bézout's theorem.

The intersection number is partly motivated by the desire to define intersection to satisfy Bézout's theorem.

Calculating the intersection numbers at the fixed points counts the fixed points with multiplicity, and leads to the Lefschetz fixed point theorem in quantitative form.

In his 1895 paper Analysis Situs, Poincaré tried to prove the theorem using topological intersection theory, which he had invented.

* Donaldson's theorem, stating that a definite intersection form of a simply connected smooth manifold of dimension 4 is diagonalisable

In fact a thorough use of Pontryagin duality here shows that the whole Kronecker theorem describes the closure of < P > as the intersection of the kernels of the χ with