Polynomial identity ring
Encyclopedia
In mathematics
, in the subfield of ring theory
, a ring R is a polynomial identity ring if there is, for some N > 0, an element P other than 0 of the free algebra
, Z1, X2, ..., XN>, over the ring of integers
in N variables X1, X2, ..., XN such that for all N-tuple
s r1, r2, ..., rN taken from R it happens that
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. The 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 pX = 0. To exclude these examples, sometimes it is defined that PI-rings must satisfy a monic polynomial identity.
PI-algebras over a field
satisfy the Kurosh conjecture, the Nullstellensatz and the catenary property
for prime ideals.
If R is a PI-ring and K is a subring of its center such that R is integral
over K then the going up and going down properties
for prime ideals of R and K are satisfied. Also the lying over property (If p is a prime ideal of K then there is a prime ideal P of R such that
) and the incomparability property (If P and Q are prime ideals of R and 
then 
) are satisfied.
1, X2, ..., XN> is the free algebra in N variables and R is a PI-ring satisfying the polynomial P in N variables, then P is in the kernel
of any homomorphism
:F 
R.
An ideal I of F is called T-ideal if
for every endomorphism
f of F.
Given a PI-ring, R, the set of all polynomial identities it satisfies is an ideal
but even more it is a T-ideal. Conversely, if I is a T-ideal of F then F/I is a PI-ring satisfying all identities in I. It is assumed that I contains monic polynomials when PI-rings are required to satisfy monic polynomial identities.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the subfield of ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, a ring R is a polynomial identity ring if there is, for some N > 0, an element P other than 0 of the free algebra
Free algebra
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring .-Definition:...
, Z
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
in N variables X1, X2, ..., XN such that for all N-tuple
Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...
s r1, r2, ..., rN taken from R it happens that
Strictly the Xi here are "non-commuting indeterminates", and so "polynomial identity" is a slight abuse of language
Abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition . Abuse of notation should be contrasted with misuse of notation, which should be avoided...
, 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. The PI-rings are usually taken as close generalizations of commutative rings. If the ring has characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
p different from zero then it satisfies the polynomial pX = 0. To exclude these examples, sometimes it is defined that PI-rings must satisfy a monic polynomial identity.
Examples
- For example if R is a commutative ringCommutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
it is a PI-ring: this is true with
-
- P(X1, X2) = X1X2 − X2X1.
- 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). It derives from the Leibniz formula for determinants
- by replacing each product in the summand by the product of the Xi in the order given by the permutation σ. In other words each of the N! orders is summed, and the coefficient is 1 or −1 according to the signature.
- The k×k matrix ringMatrix ringIn abstract algebra, a matrix ring is any collection of matrices forming a ring under matrix addition and matrix multiplication. The set of n×n matrices with entries from another ring is a matrix ring, as well as some subsets of infinite matrices which form infinite matrix rings...
over any commutative ring satisfies a standard identity; the Amitsur–Levitzki theorem states that it satisfies s2k. The degree of this identity is optimal since the matrix ring can not satisfy any monic polynomial of degree less than 2k.
- Given a field k of characteristic zero, take R to be the exterior algebraExterior algebraIn mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...
over a countably infinite-dimensional vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
with basis e1, e2, e3, ... Then R is generated by the elements of this basis and
-
- eiej = −ejei.
- This ring does not satisfy sN for any N and therefore can not be embedded in any matrix ring. In fact sN(e1,e2,...,eN) = N!e1e2...eN ≠ 0. On the other hand it is a PI-ring since it satisfies , y], z] := xyz − xzz − zxy + zyz = 0. It is enough to check this for monomials in the es. Now, a monomial of even degree commutes with every element. Therefore if either x or y is a monomial of even degree [x, y] := xy − yx = 0. If both are of odd degree then [x, y] = xy − yx = 2xy has even degree and therefore commutes with z, i.e. , y], z] = 0.
Properties
- Any subringSubringIn mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...
or homomorphic imageHomomorphismIn abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
of a PI-ring is a PI-ring. - A finite direct productDirect productIn mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
of PI-rings is a PI-ring. - A direct product of PI-rings, satisfying the same identity, is a PI-ring.
- It can always be assumed that the identity that the PI-ring satisfies is multilinear.
- If a ring is finitely generated by n elements as a moduleModule (mathematics)In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over its centerCenter (algebra)The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. It is often denoted Z, from German Zentrum, meaning "center". More specifically:...
then it satisfies every alternating multilinear polynomial of degree larger than n. In particular it satisfies sN for N > n and therefore it is a PI-ring. - If R and S are PI-rings then their tensor productTensor productIn mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
over the integers,
, is also a PI-ring.
PI-rings as generalizations of commutative rings
Among noncommutative rings, PI-rings satisfy the Köthe conjecture. AffineFinitely generated algebra
In mathematics, a finitely generated algebra is an associative algebra A over a field K where there exists a finite set of elements a1,…,an of A such that every element of A can be expressed as a polynomial in a1,…,an, with coefficients in K...
PI-algebras over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
satisfy the Kurosh conjecture, the Nullstellensatz and the catenary property
Catenary ring
In mathematics, a commutative ring R is catenary if for any pair of prime idealsany two strictly increasing chainsare contained in maximal strictly increasing chains from p to q of the same length. In other words, there is a well-defined function from pairs of prime ideals to natural numbers,...
for prime ideals.
If R is a PI-ring and K is a subring of its center such that R is integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
over K then the going up and going down properties
Going up and going down
In commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain properties of chains of prime ideals in integral extensions....
for prime ideals of R and K are satisfied. Also the lying over property (If p is a prime ideal of K then there is a prime ideal P of R such that
) and the incomparability property (If P and Q are prime ideals of R and then ) are satisfied.The set of identities a PI-ring satisfies
If F := ZKernel (mathematics)
In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element , as in kernel of a linear operator and kernel of a matrix...
of any homomorphism
:F R.An ideal I of F is called T-ideal if
for every endomorphismEndomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
f of F.
Given a PI-ring, R, the set of all polynomial identities it satisfies is an ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
but even more it is a T-ideal. Conversely, if I is a T-ideal of F then F/I is a PI-ring satisfying all identities in I. It is assumed that I contains monic polynomials when PI-rings are required to satisfy monic polynomial identities.