Center (algebra)

Center (algebra)

The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. More specifically:
* The center of a group "G" consists of all those elements "x" in "G" such that "xg" = "gx" for all "g" in "G". This is a normal subgroup of "G".
* The center of a ring "R" is the subset of "R" consisting of all those elements "x" of "R" such that "xr" = "rx" for all "r" in "R". The center is a commutative subring of "R", so "R" is an algebra over its center.
* The center of an algebra "A" consists of all those elements "x" of "A" such that "xa" = "ax" for all "a" in "A". See also: central simple algebra.
* The center of a Lie algebra "L" consists of all those elements "x" in "L" such that ["x","a"] = 0 for all "a" in "L". This is an ideal of the Lie algebra "L".
* The center of a monoidal category C consists of pairs "(A,u)" where "A" is an object of C, and u:A otimes - ightarrow - otimes A a natural isomorphism satisfying certain axioms.


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Center — or centre (see American and British English spelling differences) may refer to: cience*Center (algebra), uses of center in algebra *Center of gravity (military) *Center (group theory), in abstract algebra, the subgroup consisting of those… …   Wikipedia

  • Center (group theory) — In abstract algebra, the center of a group G is the set Z ( G ) of all elements in G which commute with all the elements of G . That is,:Z(G) = {z in G | gz = zg ;forall,g in G}.Note that Z ( G ) is a subgroup of G , because # Z ( G ) contains e …   Wikipedia

  • Algebra (ring theory) — In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R .Any ring can be thought of as an algebra… …   Wikipedia

  • Algebra over a field — This article is about a particular kind of vector space. For other uses of the term algebra , see algebra (disambiguation). In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it is… …   Wikipedia

  • Center for Information Technology (Specialty Center) — The Center for Information Technology (CIT) is a Henrico County Public Schools specialty center, located at Deep Run High School, that prepares students for higher education and careers in related fields of computer science and information… …   Wikipedia

  • Center for Advanced Technologies — The Center for Advanced Technologies (CAT) is a public magnet program in St. Petersburg, Florida, attached to Lakewood High School and part of the Pinellas County Schools district. Its primary focus is mathematics, science, and technology. It is… …   Wikipedia

  • List of abstract algebra topics — Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this …   Wikipedia

  • Axiom (computer algebra system) — Scratchpad redirects here. For scratchpad memory, see Scratchpad RAM. Axiom Developer(s) independent group of people Stable release September 2011 Operating system cross platform …   Wikipedia

  • Division algebra — In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible. Contents 1 Definitions 2 Associative division algebras 3 Not necessarily asso …   Wikipedia

  • Von Neumann algebra — 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. They were originally introduced by John von Neumann,… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”