# 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.

## Definitions and major results

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

The category of commutative rings has

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

## 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 ${\displaystyle A}$, 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 ${\displaystyle A}$, primary decomposition of an ideal ${\displaystyle I}$ translates exactly into a decomposition of the closed subset ${\displaystyle V(I)}$ into irreducible components.

Formally speaking, the assignment of a ring ${\displaystyle A}$ to its prime spectrum ${\displaystyle Spec(A)}$ 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.