proof of Cayley-Hamilton theorem in a commutative ring
Let be a commutative ring with identity and let be an order matrix with elements from . For example, if is
In this way we have a mapping which is an isomorphism of the rings and .
Now view this as an equation in . It says that is a left factor of . So by the factor theorem, the left hand value of at is 0. The coefficients of have the form , for , so they commute with . Therefore right and left hand values are the same.
- 1 Malcom F. Smiley. Algebra of Matrices. Allyn and Bacon, Inc., 1965. Boston, Mass.
|Title||proof of Cayley-Hamilton theorem in a commutative ring|
|Date of creation||2013-03-22 16:03:16|
|Last modified on||2013-03-22 16:03:16|
|Last modified by||Mathprof (13753)|