Ring (mathematics)In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
Center (ring theory)In algebra, the center of a ring R is the subring consisting of the elements x such that xy = yx for all elements y in R. It is a commutative ring and is denoted as ; "Z" stands for the German word Zentrum, meaning "center". If R is a ring, then R is an associative algebra over its center. Conversely, if R is an associative algebra over a commutative subring S, then S is a subring of the center of R, and if S happens to be the center of R, then the algebra R is called a central algebra.
Quotient ringIn ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring R and a two-sided ideal I in R, a new ring, the quotient ring R / I, is constructed, whose elements are the cosets of I in R subject to special + and ⋅ operations.
Rings of SaturnThe rings of Saturn are the most extensive ring system of any planet in the Solar System. They consist of countless small particles, ranging in size from micrometers to meters, that orbit around Saturn. The ring particles are made almost entirely of water ice, with a trace component of rocky material. There is still no consensus as to their mechanism of formation. Although theoretical models indicated that the rings were likely to have formed early in the Solar System's history, newer data from Cassini suggested they formed relatively late.
Principal component analysisPrincipal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and enabling the visualization of multidimensional data. Formally, PCA is a statistical technique for reducing the dimensionality of a dataset. This is accomplished by linearly transforming the data into a new coordinate system where (most of) the variation in the data can be described with fewer dimensions than the initial data.
Local ringIn mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal.
Valuation ringIn abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x−1 belongs to D. Given a field F, if D is a subring of F such that either x or x−1 belongs to D for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring.
Artinian ringIn mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields.
TensorIn mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product.
Ring homomorphismIn ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function f : R → S such that f is: addition preserving: for all a and b in R, multiplication preserving: for all a and b in R, and unit (multiplicative identity) preserving: Additive inverses and the additive identity are part of the structure too, but it is not necessary to require explicitly that they too are respected, because these conditions are consequences of the three conditions above.
