View source for PI-algebra

An algebra over a field for which certain polynomial identities are true.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p0726401.png" /> be an associative algebra (cf. [[Associative rings and algebras|Associative rings and algebras]]) over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p0726402.png" />, let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p0726403.png" /></td> </tr></table>

be the [[Free associative algebra|free associative algebra]] (the algebra of non-commutative polynomials) on a countable set of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p0726404.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p0726405.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p0726406.png" /> be a non-zero element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p0726407.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p0726408.png" /></td> </tr></table>

is said to be a polynomial identity of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p0726409.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264010.png" /> for every choice of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264011.png" />.

Examples of PI-algebras and of identities. The following identity is true in a [[Commutative algebra|commutative algebra]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264012.png" /></td> </tr></table>

(identity of commutativity); in the [[Exterior algebra|exterior algebra]] of a linear space the metAbelian identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264013.png" /> is satisfied; an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264014.png" /> of finite dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264015.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264016.png" /> satisfies the so-called standard identity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264017.png" />-th degree

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264018.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264019.png" /> is the group of permutations of the set consisting of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264020.png" /> positive integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264021.png" />; it also satisfies the more general Capelli identity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264022.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264023.png" /></td> </tr></table>

In the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264024.png" /> of square matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264025.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264026.png" /> the standard identity of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264027.png" /> is satisfied. A tensor product of PI-algebras is a PI-algebra.

For any PI-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264028.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264029.png" /> of characteristic zero it is possible to find a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264030.png" /> such that the identities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264031.png" /> are implied by the powers of the identities of the matrix algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264032.png" />; moreover, some power of any identity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264033.png" /> is an identity of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264034.png" />. Thus, in any PI-algebra over a field of characteristic zero some power of the standard identity is satisfied.

The totality of all left-hand sides of the identities which are satisfied in a given algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264035.png" /> forms a fully characteristic ideal ([[T-ideal|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264036.png" />-ideal]] for short) of the free algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264037.png" />; conversely, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264038.png" />-ideal there exists an algebra whose set of identities coincides with this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264039.png" />-ideal (for example, the quotient algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264040.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264041.png" /> is of characteristic zero, the identities can be differentiated, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264042.png" />-ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264043.png" /> are precisely the differentially closed one-sided ideals. For instance, repeated differentiation of the nil identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264044.png" /> yields the identity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264045.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264046.png" /></td> </tr></table>

which is multi-linear (or, more exactly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264048.png" />-linear), i.e. linear with respect to each one of its constituent variables. Conversely, setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264049.png" /> in the last identity one obtains the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264050.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264051.png" />. This process of linearization of identities makes it possible to state (for fields of characteristic zero) that all the identities of the algebra are consequences of its multi-linear identities. For an algebra with unit element, moreover, all its identities result from those of its multi-linear identities which are representable by linear combinations of products of right-normalized commutators (cf. [[Commutator|Commutator]]) of different degrees in the generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264052.png" />. The Specht problem deals with the question of whether all associative algebras have a finite basis for the identities.

The totality of all algebras which satisfy a given system of identities is called a variety. A variety may also be defined as a class of algebras closed with respect to taking subalgebras, homomorphic images and subdirect products (cf. also [[Algebraic systems, variety of|Algebraic systems, variety of]]). A number of varieties of algebras have been demonstrated to be finitely based (i.e. Specht's problem has a positive solution in such varieties). Such varieties include those (again over a field of characteristic zero) of nilpotent algebras of a given index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264053.png" />, algebras in which the additive commutators of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264054.png" /> are zero (Lie-nilpotent algebras), and the variety of algebras defined by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264055.png" />-ideal of semi-identities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264056.png" /> (the algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264057.png" />-matrices). However, the problem remains open for the variety defined by an ideal of identities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264058.png" />, i.e. for the matrix algebras of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264059.png" />.

The existence of a polynomial identity rigidly determines the structure of an associative algebra. A primitive algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264060.png" /> (cf. [[Primitive ring|Primitive ring]]) which satisfies a polynomial identity of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264061.png" /> is isomorphic to a matrix algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264062.png" /> over a [[Skew-field|skew-field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264063.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264064.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264065.png" /></td> </tr></table>

Accordingly, a semi-simple (in the sense of the [[Jacobson radical|Jacobson radical]]) PI-algebra can be expanded into a subdirect sum of complete matrix algebras over skew-fields, the orders of the matrix algebras and the dimensions of the skew-fields over the centres being bounded in the set, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264066.png" />-ideal of identities of the semi-simple algebra coinciding with some "matrix" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264067.png" />-ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264068.png" />. An ordered PI-algebra is commutative. A primary PI-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264069.png" /> (cf. [[Primary ring|Primary ring]]) has a two-sided classical quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264070.png" />, which is isomorphic to a matrix algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264071.png" /> over a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264072.png" />, the latter being finite-dimensional over its centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264073.png" />. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264074.png" /> is a central extension of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264075.png" /> in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264076.png" />. The ideals of identities of the algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264078.png" /> are the same. PI-algebras satisfy a number of conditions of Burnside type (cf. [[Burnside problem|Burnside problem]]). For instance, an algebraic (nil) PI-algebra is locally finite (locally nilpotent). An associative [[Nil algebra|nil algebra]] of bounded index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264079.png" /> is nilpotent if the characteristic of the ground field is zero or larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264080.png" />.

A PI-algebra without non-zero nil ideals is representable by matrices over a commutative ring. However, not all PI-algebras are representable in this way. For example, the exterior algebra of a countably-dimensional space is not so representable, since it does not satisfy any standard identity. The internal characterization of the representability of an algebra by matrices over a commutative ring is an independent branch of study in the theory of PI-algebras.

The Jacobson radical of a finitely-generated PI-algebra over a field of characteristic zero is a nil ideal. At the time of writing (1977) the question of its nilpotency is still open. If the Jacobson radical of a PI-algebra is nilpotent, this algebra satisfies all the identities of a matrix algebra of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264081.png" /> for some value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264082.png" />. The converse proposition has been demonstrated for finitely-generated algebras. Moreover, for a finitely-generated algebra over a field of characteristic zero, nilpotency of the Jacobson radical is equivalent to the validity of some standard identity in this algebra.

If an identity is satisfied for "a part" of the elements of an algebra, it follows in many cases that some identity is satisfied for all elements of the algebra. For instance, if the symmetric elements in an algebra with involution (cf. [[Involution algebra|Involution algebra]]) satisfy an identity, the algebra is a PI-algebra; if a finite group of automorphisms acts on an algebra over a field of characteristic zero and if an invariant subalgebra satisfies a given identity, the initial algebra will be a PI-algebra.

It is interesting to inquire into the conditions under which given special algebras satisfy a polynomial identity.

For the [[Group algebra|group algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264083.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264084.png" /> over a field of characteristic zero to satisfy some polynomial identity it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264085.png" /> have an Abelian subgroup of finite index. If the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264086.png" /> is finite and equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264088.png" /> is a PI-algebra if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264089.png" /> has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264090.png" />-Abelian subgroup of finite index (a group is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264092.png" />-Abelian if its commutator is a finite [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264093.png" />-group]]).

The [[Universal enveloping algebra|universal enveloping algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264094.png" /> of a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264095.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264096.png" /> of characteristic zero is a PI-algebra if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264097.png" /> is Abelian (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264098.png" /> is commutative). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p07264099.png" /> is a field of a finite characteristic, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640100.png" /> will be a PI-algebra if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640101.png" /> has an Abelian ideal of finite codimension while the adjoint representation of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640102.png" /> is of bounded algebraic degree.

All PI-subalgebras of a free associative algebra are commutative.

The theory of PI-algebras is a natural extension of commutative algebra. It contains deep and decisive analogues of theorems in commutative algebra, so that one may speak of the construction of non-commutative algebraic geometry.

Any finitely-generated PI-algebra with generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640103.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640104.png" /> satisfies the condition of boundedness of heights, i.e. there exists a finite number of words <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640105.png" /> in the generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640106.png" /> and a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640107.png" /> such that any word <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640108.png" /> in the generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640109.png" /> is representable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640110.png" /> by a linear combination of the words

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640111.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640112.png" />, the composition of which with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640113.png" /> coincides with the word <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640114.png" />. In the commutative case the generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640115.png" /> themselves may be used as the words <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640116.png" />. A free non-commutative affine ring is a quotient algebra

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640117.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640118.png" /> is a free algebra with a finite number of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640119.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640120.png" /> of characteristic zero while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640121.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640122.png" />-ideal of identities of the matrix algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640123.png" /> as defined above. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640124.png" /> is a PI-algebra without divisors of zero, and has classical skew-field of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640125.png" /> which is finite-dimensional over its centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640126.png" />. Further, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640127.png" /> be a space whose elements are columns of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640128.png" />, constituted by matrices of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640129.png" />. One may speak of the zeros of the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640130.png" /> that are located in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640131.png" />, of algebraic varieties in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640132.png" />, etc., the fundamental assumptions of classical algebraic geometry being satisfied. Thus, the non-commutative analogue of Hilbert's Nullstellensatz (cf. [[Hilbert theorem|Hilbert theorem]]) is satisfied. Primary ideals of the algebra which meet the condition of being Noetherian correspond to irreducible algebraic varieties. Krull's theorem describing the coincidence of the maximum chain length of primary ideals of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640133.png" /> with the degree of transcendence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640134.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640135.png" />, which, in the present case, is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640136.png" /></td> </tr></table>

is satisfied.

By analogy with associative algebras it is possible to define, using the elements of free algebras, PI-algebras in other classes of algebras comprising free algebras (Lie algebras, alternative algebras, etc.).

A Lie algebra over a field of characteristic zero which satisfies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640137.png" />-th Engel identity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640138.png" /></td> </tr></table>

is locally nilpotent. Higgins' problem (namely, does Engel's identity imply nilpotency?) has been positively solved for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072640/p072640139.png" /> only. Its solution is negative for fields of positive characteristic.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Procesi, "Rings with polynomial identities" , M. Dekker (1973) {{MR|0366968}} {{ZBL|0262.16018}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) {{MR|0081264}} {{ZBL|0073.02002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) {{MR|1535024}} {{MR|0227205}} {{ZBL|0177.05801}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> P.M. Cohn, "Universal algebra" , Reidel (1981) {{MR|0620952}} {{ZBL|0461.08001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.I. Shirshov, "On rings with identity relations" ''Mat. Sb.'' , '''43 (85)''' : 2 (1957) pp. 277–283 (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A.I. Kostrikin, "On Burnside's problem" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''3''' : 1 (1959) pp. 3–34 (In Russian) {{MR|132100}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> G. Higman, "On a conjecture of Nagata" ''Proc. Cambridge Philos. Soc. (1)'' , '''52''' (1956) pp. 1–4 {{MR|0073581}} {{ZBL|0072.02502}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> P.J. Higgins, "Lie rings satisfying the Engel condition" ''Proc. Cambridge Philos. Soc. (1)'' , '''50''' (1954) pp. 8–15 {{MR|0059890}} {{ZBL|0055.02601}} </TD></TR></table>



====Comments====
A.R. Kemer has shown that every variety of associative algebras over a field of characteristic zero is finitely based [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.R. Kemer, "Solution of the problem as to whether associative algebras have a finite basis of identities" ''Soviet Math. Dokl.'' , '''37''' (1988) pp. 60–64 ''Dokl. Akad. Nauk SSSR'' , '''298''' (1988) pp. 273–277 {{MR|0937115}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L.H. Rowen, "Polynomial identities in ring theory" , Acad. Press (1980) pp. Chapt. 7 {{MR|0576061}} {{ZBL|0461.16001}} </TD></TR></table>