Separable algebra

Separable algebra

A separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension.

Definition

Let "K" be a field. An associative "K"-algebra "A" is said to be separable if for every field extension scriptstyle L/K the algebra scriptstyle Aotimes_K L is semisimple.

Commutative separable algebras

If scriptstyle L/K is a field extension, then "L" is separable as an associative "K"-algebra if and only if the extension of field is separable.

Examples

If "K" is a field and "G" is a finite group such that the order of "G" is invertible in "K", then the group ring "K" ["G"] is a separable "K"-algebra.

References

* Charles Weibel, "Homological algebra"


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • separable Körpererweiterung — separable Körper|erweiterung,   Algebra: ein Erweiterungskörper L des Körpers K, in dem jedes Element a ∈ K separabel ist (separables Element) …   Universal-Lexikon

  • Separable polynomial — In mathematics, two slightly different notions of separable polynomial are used, by different authors. According to the most common one, a polynomial P(X) over a given field K is separable if all its roots are distinct in an algebraic closure of… …   Wikipedia

  • Separable sigma algebra — In mathematics, sigma; algebras are usually studied in the context of measure theory. A separable sigma; algebra (or separable sigma; field) is a sigma algebra that can be generated by a countable collection of sets. To learn what is meant by the …   Wikipedia

  • Separable extension — In mathematics, an algebraic field extension L / K is separable if it can be generated by adjoining to K a set each of whose elements is a root of a separable polynomial over K . In that case, each beta; in L has a separable minimal polynomial… …   Wikipedia

  • Álgebra de Borel — En matemáticas, el álgebra de Borel (más correctamente, σ álgebra de Borel, también llamada boreliana) sobre un espacio topológico X es una σ álgebra de subconjuntos de X asociada a la topología de X. En la literatura matemática se pueden… …   Wikipedia Español

  • Separable σ-Algebra — In der Maßtheorie wird eine σ Algebra als separabel bezeichnet, wenn sie aus einer abzählbaren Anzahl von Mengen erzeugt werden kann. Beispiel Die Borel schen σ Algebren sind separabel, denn sie werden von den Quadern mit rationalen Endpunkten… …   Deutsch 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

  • Spectrum of a C*-algebra — The spectrum of a C* algebra or dual of a C* algebra A, denoted Â, is the set of unitary equivalence classes of irreducible * representations of A. A * representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed… …   Wikipedia

  • Abelian von Neumann algebra — In functional analysis, an Abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute. The prototypical example of an abelian von Neumann algebra is the algebra L^infty(X,mu) for μ a σ… …   Wikipedia

  • C*-algebra — C* algebras (pronounced C star ) are an important area of research in functional analysis, a branch of mathematics. The prototypical example of a C* algebra is a complex algebra A of linear operators on a complex Hilbert space with two additional …   Wikipedia

Share the article and excerpts

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