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In [[ring theory]], a branch of [[mathematics]], a [[ring (mathematics)|ring]] ''R'' is a '''polynomial identity ring''' if there is, for some ''N'' > 0, an element ''P'' ≠ 0 of the [[free algebra]], '''Z'''{{angle bracket|''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''N''</sub>}}, over the ring of [[Integer#Algebraic_properties|integers]] in ''N'' variables ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''N''</sub> such that
:<math>P(r_1, r_2, \ldots, r_N) = 0</math>
for all ''N''-[[tuple]]s ''r''<sub>1</sub>, ''r''<sub>2</sub>, ..., ''r''<sub>''N''</sub> taken from ''R''.

Strictly the ''X''<sub>''i''</sub> here are "non-commuting indeterminates", and so "polynomial identity" is a slight [[abuse of notation|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 a polynomial|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 (algebra)|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.<ref>J.C. McConnell, J.C. Robson, ''Noncommutative Noetherian Rings, [[Graduate Studies in Mathematics]], Vol 30''</ref>

==Examples==
* For example, if ''R'' is a [[commutative ring]] it is a PI-ring: this is true with
::<math>P(X_1,X_2) = X_1X_2-X_2X_1 = 0~</math>

*The ring of 2&thinsp;×&thinsp;2 [[matrix (mathematics)|matrices]] over a commutative ring satisfies the '''Hall identity'''
::<math>(xy-yx)^2z=z(xy-yx)^2</math>
:This identity was used by {{harvs|txt|last=Hall|first=M.|year=1943}}, but was found earlier by {{harvs|txt|last=Wagner|year=1937}}.
* A major role is played in the theory by the '''standard identity''' ''s''<sub>''N''</sub>, of length ''N'', which generalises the example given for commutative rings (''N'' = 2). It derives from the [[Leibniz formula for determinants]]

::<math>\det(A) = \sum_{\sigma \in S_N} \sgn(\sigma) \prod_{i = 1}^N a_{i,\sigma(i)}</math>

:by replacing each product in the summand by the product of the ''X''<sub>''i''</sub> in the order given by the [[permutation]] σ. In other words each of the ''N''&hairsp;! orders is summed, and the coefficient is 1 or −1 according to the [[signature of a permutation|signature]].

::<math>s_N(X_1,\ldots,X_N) = \sum_{\sigma \in S_N} \sgn(\sigma) X_{\sigma(1)}\dotsm X_{\sigma(N)}=0~</math>

:The ''m''&thinsp;×&thinsp;''m'' [[matrix ring]] over any commutative ring satisfies a standard identity: the [[Amitsur–Levitzki theorem]] states that it satisfies ''s''<sub>2''m''</sub>. The degree of this identity is optimal since the matrix ring can not satisfy any monic polynomial of degree less than&nbsp;2''m''.

* Given a [[field (mathematics)|field]] ''k'' of characteristic zero, take ''R'' to be the [[exterior algebra]] over a [[countably infinite]]-[[dimension (vector space)|dimensional]] [[vector space]] with [[basis (linear algebra)|basis]] ''e''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>3</sub>, ... Then ''R'' is generated by the elements of this basis and

::''e''<sub>''i''</sub>&thinsp;''e''<sub>''j''</sub> = −&hairsp;''e''<sub>''j''</sub>&thinsp;''e''<sub>''i''</sub>.

:This ring does not satisfy ''s''<sub>''N''</sub> for any ''N'' and therefore can not be [[embedding|embedded]] in any matrix ring. In fact ''s''<sub>''N''</sub>(''e''<sub>1</sub>,''e''<sub>2</sub>,...,''e''<sub>''N''</sub>) =&nbsp;''N''&hairsp;!&hairsp;''e''<sub>1</sub>''e''<sub>2</sub>...''e''<sub>''N''</sub>&nbsp;≠&nbsp;0. On the other hand it is a PI-ring since it satisfies [[''x'',&nbsp;''y''],&nbsp;''z'']&nbsp;:=&nbsp;''xyz''&nbsp;−&nbsp;''yxz''&nbsp;−&nbsp;''zxy''&nbsp;+&nbsp;''zyx''&nbsp;=&nbsp;0. It is enough to check this for monomials in the ''e''<sub>''i''</sub>'s. Now, a monomial of [[parity (mathematics)|even]] degree commutes with every element. Therefore if either ''x'' or ''y'' is a monomial of even degree [''x'',&nbsp;''y''] :=&nbsp;''xy''&nbsp;−&nbsp;''yx''&nbsp;=&nbsp;0. If both are of [[parity (mathematics)|odd]] degree then [''x'',&nbsp;''y'']&nbsp;=&nbsp;''xy''&nbsp;−&nbsp;''yx''&nbsp;=&nbsp;2''xy'' has even degree and therefore commutes with ''z'', i.e. [[''x'',&nbsp;''y''],&nbsp;''z'']&nbsp;=&nbsp;0.

==Properties==
* Any [[subring]] or [[ring homomorphism|homomorphic]] [[image (mathematics)|image]] of a PI-ring is a PI-ring.
* A finite [[product ring|direct product]] 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 form|multilinear]].
* If a ring is [[finitely generated module|finitely generated]] by ''n'' elements as a [[module (mathematics)|module]] over its [[center (ring theory)|center]] then it satisfies every alternating multilinear polynomial of degree larger than ''n''. In particular it satisfies ''s''<sub>''N''</sub> for ''N''&nbsp;>&thinsp;''n'' and therefore it is a PI-ring.
* If ''R'' and ''S'' are PI-rings then their [[tensor product]] over the integers, <math>R\otimes_\mathbb{Z}S</math>, is also a PI-ring.
* If ''R'' is a PI-ring, then so is the ring of ''n''&thinsp;×&thinsp;''n'' matrices with coefficients in ''R''.

==PI-rings as generalizations of commutative rings==
Among non-commutative rings, PI-rings satisfy the [[Köthe conjecture]]. [[Finitely generated algebra|Affine]] PI-algebras over a [[field (mathematics)|field]] satisfy the [[Kurosh conjecture]], the [[Nullstellensatz]] and the [[catenary ring|catenary property]] for [[prime ideal]]s.

If ''R'' is a PI-ring and ''K'' is a subring of its center such that ''R'' is [[integral extension|integral over]] ''K'' then the [[Going up and going down|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 <math>p</math> is minimal over <math>P\cap K</math>) and the ''incomparability'' property (If ''P'' and ''Q'' are prime ideals of ''R'' and <math>P\subset Q</math> then <math>P\cap K\subset Q\cap K</math>) are satisfied.
<!--Add here results about integral extensions similar to commutative case.-->

==The set of identities a PI-ring satisfies==

If ''F''&nbsp;:=&nbsp;'''Z'''{{angle bracket|''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''N''</sub>}} 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 (algebra)|kernel]] of any homomorphism
:<math>\tau</math>: ''F'' <math>\rightarrow</math> ''R''.

An [[ideal (ring theory)|ideal]] ''I'' of ''F'' is called '''T-ideal''' if <math>f(I)\subset I</math> for every [[endomorphism]] ''f'' of ''F''.

Given a PI-ring, ''R'', the set of all polynomial identities it satisfies is an [[Ideal (ring theory)|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.

==See also==
*[[Posner's theorem]]
*[[Central polynomial]]

==References==
{{Reflist}}
*{{eom|id=P/p072640|title=PI-algebra|first=V.N.|last= Latyshev}}
*{{eom|id=a/a110570|title=Amitsur–Levitzki theorem |first=E.|last= Formanek|authorlink= Edward W. Formanek}}
*[https://books.google.com/books?id=Li147JZ4T6AC Polynomial identities in ring theory], Louis Halle Rowen, Academic Press, 1980, {{ISBN|978-0-12-599850-5}}
*[https://books.google.com/books?id=x8gJw5bMw2oC Polynomial identity rings], Vesselin S. Drensky, Edward Formanek, Birkhäuser, 2004, {{ISBN|978-3-7643-7126-5}}
*[https://books.google.com/books?id=ZLW_Kz_zOP8C Polynomial identities and asymptotic methods], A. Giambruno, Mikhail Zaicev, AMS Bookstore, 2005, {{ISBN|978-0-8218-3829-7}}
*[https://books.google.com/books?id=80pw1QoLSQUC Computational aspects of polynomial identities], Alexei Kanel-Belov, Louis Halle Rowen, A K Peters Ltd., 2005, {{ISBN|978-1-56881-163-5}}

==Further reading==
* {{cite book | last=Formanek | first=Edward | title=The polynomial identities and invariants of ''n''×''n'' matrices | zbl=0714.16001 | series=Regional Conference Series in Mathematics | volume=78 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1991 | isbn=0-8218-0730-7 }}
* {{cite book | last1=Kanel-Belov | first1=Alexei | last2=Rowen | first2=Louis Halle | title=Computational aspects of polynomial identities | zbl=1076.16018 | series=Research Notes in Mathematics | volume=9 | location=Wellesley, MA | publisher=A K Peters | isbn=1-56881-163-2 | year=2005 }}

==External links==
*{{PlanetMath|urlname=PolynomialIdentityAlgebra|title=Polynomial identity algebra}}
*{{PlanetMath|urlname=StandardIdentity|title=Standard Identity}}
*{{PlanetMath|urlname=TIdeal|title=T-ideal}}

{{DEFAULTSORT:Polynomial Identity Ring}}
[[Category:Ring theory]]