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In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced, which are often denoted with the Hebrew letter (aleph) marked with subscript indicating their rank among the infinite cardinals.
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.
Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
In ring theory, a branch of mathematics, a radical of a ring is an ideal of "not-good" elements of the ring.
In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero.
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of generators of m. Then by Krull's principal ideal theorem n ≥ dim A, and A is defined to be regular if n = dim A.
In mathematics, a von Neumann regular ring is a ring R such that for every element a in R there exists an x in R with a = axa. One may think of x as a "weak inverse" of the element a; in general x is not uniquely determined by a. Von Neumann regular rings are also called absolutely flat rings, because these rings are characterized by the fact that every left R-module is flat.
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings. An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings.
In mathematics, a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there exists a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite. Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers.
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like R, S are Morita equivalent if their categories of modules are additively equivalent. It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.
In mathematics, especially in the area of abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective. If this is required only for finitely generated submodules, it is called semihereditary.
In mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. When this is satisfied for both left and right ideals, such as the case when R is a commutative ring, R can be called a principal ideal ring, or simply principal ring.
In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer.
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Presently, the term "abstract algebra" is typically used for naming courses in mathematical education, and is rarely used in advanced mathematics.
In mathematics, especially ring theory, a regular ideal can refer to multiple concepts.
In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) R-module M has a singular submodule consisting of elements whose annihilators are essential right (resp. left) ideals in R. In set notation it is usually denoted as . For general rings, is a good generalization of the torsion submodule tors(M) which is most often defined for domains. In the case that R is a commutative domain, .
In mathematics, continuous geometry is an analogue of complex projective geometry introduced by von Neumann, where instead of the dimension of a subspace being in a discrete set , it can be an element of the unit interval . Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.
References
- 1 2 Goodearl, Kenneth (1976). Ring Theory: Nonsingular Rings and Modules. CRC Press. pp. 165–166. ISBN 978-0-8247-6354-1.
- 1 2 3 4 5 6 Lam, T. Y. (2012-12-06). A First Course in Noncommutative Rings. Springer Science & Business Media. ISBN 978-1-4684-0406-7.
