Polynomial identity ring

In ring theory, a branch of mathematics, a ring R is a polynomial identity ring if there is, for some N > 0, an element P ≠ 0 of the free algebra, Z⟨X1, X2, ..., XN⟩, over the ring of integers in N variables X1, X2, ..., XN such that

P

(

r

1

,

r

2

,

…

,

r

N

)

=

0

{\displaystyle P(r_{1},r_{2},\ldots ,r_{N})=0}

for all N-tuples r1, r2, ..., rN taken from R.

Strictly the Xi here are "non-commuting indeterminates", and so "polynomial identity" is a slight abuse of language, since "polynomial" here stands for what is usually called a "non-commutative polynomial". The abbreviation PI-ring is common. More generally, the free algebra over any ring S may be used, and gives the concept of PI-algebra.

If the degree of the polynomial P is defined in the usual way, the polynomial P is called monic if at least one of its terms of highest degree has coefficient equal to 1.

Every commutative ring is a PI-ring, satisfying the polynomial identity XY − YX = 0. Therefore, PI-rings are usually taken as close generalizations of commutative rings. If the ring has characteristic p different from zero then it satisfies the polynomial identity pX = 0. To exclude such examples, sometimes it is defined that PI-rings must satisfy a monic polynomial identity.

Examples

For example, if R is a commutative ring it is a PI-ring: this is true with

P

(

X

1

,

X

2

)

=

X

1

X

2

−

X

2

X

1

=

0

{\displaystyle P(X_{1},X_{2})=X_{1}X_{2}-X_{2}X_{1}=0~}

The ring of 2 × 2 matrices over a commutative ring satisfies the Hall identity

(

x

y

−

y

x

)

2

z

=

z

(

x

y

−

y

x

)

2

{\displaystyle (xy-yx)^{2}z=z(xy-yx)^{2}}

This identity was used by M. Hall (1943), but was found earlier by Wagner (1937).

A major role is played in the theory by the standard identity sN, of length N, which generalises the example given for commutative rings (N = 2).