In
algebra, a central polynomial for ''n''-by-''n''
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 matrices was discovered in 1970 independently by
Formanek and Razmyslov. The term "central" is because the evaluation of a central polynomial has the image lying in the
center of the
matrix ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
over any
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
. The notion has an application to the theory of
polynomial identity rings.
Example:
is a central polynomial for 2-by-2-matrices. Indeed, by the
Cayley–Hamilton theorem, one has that
for any 2-by-2-matrices ''x'' and ''y''.
See also
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Generic matrix ring In algebra, a generic matrix ring is a sort of a universal matrix ring.
Definition
We denote by F_n a generic matrix ring of size ''n'' with variables X_1, \dots X_m. It is characterized by the universal property: given a commutative ring ''R'' a ...
References
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{{algebra-stub
Ring theory