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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, Z<''X''1, ''X''2, ..., ''X''''N''>, over the ring of integers in ''N'' variables ''X''1, ''X''2, ..., ''X''''N'' such that for all ''N''-tuples ''r''1, ''r''2, ..., ''r''''N'' taken from ''R'' it happens that : Strictly the ''X''''i'' 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.〔J.C. McConnell, J.C. Robson, ''Noncommutative Noetherian Rings, Graduate Studies in Mathematics, Vol 30''〕 ==Examples== * For example if ''R'' is a commutative ring it is a PI-ring: this is true with :: *The ring of 2 by 2 matrices over a commutative ring satisfies the Hall identity :: :This identity was used by , but was found earlier by . * A major role is played in the theory by the standard identity ''s''''N'', 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 ''X''i 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 ''m''×''m'' matrix ring over any commutative ring satisfies a standard identity: the Amitsur–Levitzki theorem states that it satisfies ''s''2''m''. The degree of this identity is optimal since the matrix ring can not satisfy any monic polynomial of degree less than 2''m''. * Given a field ''k'' of characteristic zero, take ''R'' to be the exterior algebra over a countably infinite-dimensional vector space with basis ''e''1, ''e''2, ''e''3, ... Then ''R'' is generated by the elements of this basis and ::''e''''i''''e''''j'' = −''e''''j''''e''''i''. :This ring does not satisfy ''s''''N'' for any ''N'' and therefore can not be embedded in any matrix ring. In fact ''s''''N''(''e''1,''e''2,...,''e''''N'') = ''N''!''e''1''e''2...''e''''N'' ≠ 0. On the other hand it is a PI-ring since it satisfies (), ''z''] := ''xyz'' − ''yxz'' − ''zxy'' + ''zyx'' = 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 () := ''xy'' − ''yx'' = 0. If both are of odd degree then () = ''xy'' − ''yx'' = 2''xy'' has even degree and therefore commutes with ''z'', i.e. (), ''z''] = 0. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「polynomial identity ring」の詳細全文を読む スポンサード リンク
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