Commutative algebra

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Commutative algebra developed as a theory in mathematics having the aim of translating classical geometric ideas into an algebraic framework, pioneered by David Hilbert and Emmy Noether at the beginning of the 20th century.

[edit] Definitions and major results

The notion of commutative ring assumes commutativity of the multiplication operation and usually also the existence of a multiplicative identity.

The category of commutative rings has

  1. commutative rings as its objects
  2. ring homomorphisms as its morphisms; i.e., functions \phi:R\to R' such that \phi is a morphism of abelian groups (with respect to the additive structure of the rings R andR'), \phi(r_1r_2)=\phi(r_1)\phi(r_2) for all r_1,r_2\in R, and \phi(1_R)=1_{R'}.

[edit] Affine Schemes

The theory of affine schemes was initiated with the definition of the prime spectrum of a ring, the set of all prime ideals of a given ring. For curves defined by polynomial equations over a ring A, the object to consider would be the prime spectrum of a polynomial ring in sufficiently many variables modulo the ideal generated by the polynomials in question. The Zariski topology (together with a structural sheaf of rings) on this set endows a geometric structure for which many illuminating algebro-geometric correspondences manifest themselves. For example, for a noetherian ring A, primary decomposition of an ideal I translates exactly into a decomposition of the closed subset V(I) into irreducible components.

Formally speaking, the assignment of a ring A to its prime spectrum Spec(A) is functorial, and is in fact an equivalence (of categories) between the category of commutative rings and affine schemes. It is this mechanism, in addition to a number of correspondence theorems, which allows us to change between the language of algebra and geometry.

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