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Central Polynomial
In Abstract algebra, algebra, a central polynomial for ''n''-by-''n'' Matrix (mathematics), matrices is a polynomial in non-commuting variables that is non-constant but yields a scalar matrix whenever it is evaluated at ''n''-by-''n'' matrices. That such polynomials exist for any Square matrix, square matrices was discovered in 1970 independently by Edward W. Formanek, Formanek and Razmyslov. The term "central" is because the evaluation of a central polynomial has the image lying in the Center (ring theory), center of the matrix ring over any commutative ring. The notion has an application to the theory of polynomial identity rings. Example: (xy - yx)^2 is a central polynomial for 2-by-2-matrices. Indeed, by the Cayley–Hamilton theorem, one has that (xy - yx)^2 = -\det(xy - yx)I for any 2-by-2-matrices ''x'' and ''y''. See also *Generic matrix ring References * * {{algebra-stub Ring theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
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 this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
