polynomial identity algebra
Let be a commutative ring with 1. Let be a countable set of variables, and let denote the free associative algebra over . If is finite, we can also write as , where the . Because of the freeness condition on the algebra, the variables are non-commuting among themselves. However, the variables do commute with elements of . A typical element of is a polynomial over in (finite) non-commuting variables of .
Definition. Let be a -algebra and . For any , is called an evaluation of at -tuple . If the evaluation vanishes (=0) for all -tuples of , then is called a polynomial identity for .
Definition. An algebra over a commutative ring is said to be a polynomial identity algebra over , or a PI-algebra over , if there is a proper polynomial , such that is a polynomial identity for . A polynomial identity ring, or PI-ring, is a polynomial identity -algebra.
A commutative ring is a PI-ring, satisfying the polynomial .
A finite field (with elements) is a PI-ring, satisfying .
The ring of matrices over a field is a PI-ring. One can show that satisfies . This identity is called the Hall identity.
|Title||polynomial identity algebra|
|Date of creation||2013-03-22 14:20:38|
|Last modified on||2013-03-22 14:20:38|
|Last modified by||CWoo (3771)|
|Synonym||algebra with polynomial identity|