Commutative algebra

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.

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 $\phi(1_R)=1_{R'}$.

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.