of prime numbers. Integral domains, 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 domains are integral domains in which every ideal can be generated by a single element, another property shared by the integers. Euclidean domains are integral domains in which the Euclidean algorithm can be carried out. Important examples of commutative rings can be constructed as rings of polynomials and their factor rings. Summary: Euclidean domain ⊂ principal ideal domain ⊂ unique factorization domain ⊂ integral domain ⊂ commutative ring. 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 ideals 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 schemes, a generalization of algebraic varieties, which may be built from any commutative ring. More precisely, the spectrum of a commutative ring is the space of its prime ideals equipped with Zariski topology, and augmented with a 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 charts of an atlas. Noncommutative rings Noncommutative rings resemble rings of matrices in many respects. Following the model of algebraic geometry, attempts have been made recently at defining noncommutative geometry based on noncommutative rings. Noncommutative rings and associative algebras (rings that are also vector spaces) are often studied via their categories of modules. A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces. Examples of noncommutative rings are given by rings of square matrices or more generally by rings of endomorphisms of abelian groups or modules, and by monoid rings. Representation theory Representation theory is a branch of mathematics that draws heavily on non-commutative rings. It studies abstract algebraic structures by representing their elements as linear transformations of