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Ring theory
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==Commutative rings== {{Main|Commutative algebra}} A ring is called ''commutative'' if its multiplication is [[commutative]]. Commutative rings resemble familiar number systems, and various definitions for commutative rings are designed to formalize properties of the [[integer]]s. Commutative rings are also important in [[algebraic geometry]]. In commutative ring theory, numbers are often replaced by [[ideal (ring theory)|ideals]], and the definition of the [[prime ideal]] tries to capture the essence of [[prime number]]s. [[Integral domain]]s, non-trivial commutative rings where no two non-zero elements multiply to give zero, generalize another property of the integers and serve as the proper realm to study divisibility. [[Principal ideal domain]]s are integral domains in which every ideal can be generated by a single element, another property shared by the integers. [[Euclidean domain]]s are integral domains in which the [[greatest common divisor|Euclidean algorithm]] can be carried out. Important examples of commutative rings can be constructed as rings of [[polynomial]]s and their factor rings. Summary: [[Euclidean domain]] β [[principal ideal domain]] β [[unique factorization domain]] β [[integral domain]] β [[commutative ring]]. ===Algebraic geometry=== {{Main|Algebraic geometry}} [[Algebraic geometry]] is in many ways the mirror image of commutative algebra. This correspondence started with [[Hilbert's Nullstellensatz]] that establishes a one-to-one correspondence between the points of an [[algebraic variety]], and the [[maximal ideal]]s of its [[coordinate ring]]. This correspondence has been enlarged and systematized for translating (and proving) most geometrical properties of algebraic varieties into algebraic properties of associated commutative rings. [[Alexander Grothendieck]] completed this by introducing [[scheme (mathematics)|scheme]]s, a generalization of algebraic varieties, which may be built from any commutative ring. More precisely, the [[spectrum of a ring|spectrum]] of a commutative ring is the space of its prime ideals equipped with [[Zariski topology]], and augmented with a [[sheaf (mathematics)|sheaf]] of rings. These objects are the "affine schemes" (generalization of [[affine varieties]]), and a general scheme is then obtained by "gluing together" (by purely algebraic methods) several such affine schemes, in analogy to the way of constructing a [[manifold]] by gluing together the [[chart (topology)|charts]] of an [[atlas (topology)|atlas]].
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