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''of a free associative algebra''
 
''of a free associative algebra''
  
A totally invariant ideal, that is, an ideal invariant under all endomorphisms. The set of all polynomial identities of an arbitrary variety of associative algebras over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t0920203.png" /> (cf. [[Associative rings and algebras|Associative rings and algebras]]) forms a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t0920204.png" />-ideal in the countably-generated free algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t0920205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t0920206.png" />. Thus, there exists a one-to-one correspondence between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t0920207.png" />-ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t0920208.png" /> and the varieties of associative algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t0920209.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202010.png" /> has characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202011.png" />, then for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202012.png" />-ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202013.png" /> there exists a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202014.png" /> such that certain powers of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202015.png" /> are elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202016.png" />, and only they, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202017.png" /> is the ideal of identities of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202018.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202019.png" />-matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202020.png" />. In this case a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202021.png" />-ideal can also be defined as a (two-sided) ideal that is closed under all differentiations of the free algebra. The quotient algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202022.png" /> is a [[PI-algebra|PI-algebra]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202023.png" /> as set of polynomial identities. It is called the relatively free algebra (or generic algebra) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202024.png" />-ideal of identities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202025.png" /> (and is a free algebra in the variety of algebras defined by the identities in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202026.png" />). The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202027.png" /> has no zero divisors if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202028.png" /> for some natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202029.png" />. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202030.png" />-ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202031.png" /> of a free associative algebra is primary.
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A totally invariant ideal, that is, an ideal invariant under all endomorphisms. The set of all polynomial identities of an arbitrary variety of associative algebras over a field $F$ (cf. [[Associative rings and algebras|Associative rings and algebras]]) forms a $T$-ideal in the countably-generated free algebra $F[X]$, $X=\{x_1,\dots,x_k,\dots\}$. Thus, there exists a one-to-one correspondence between the $T$-ideals of $F[X]$ and the varieties of associative algebras over $F$. If $F$ has characteristic $0$, then for every $T$-ideal $T\subseteq F[X]$ there exists a natural number $n=n(T)$ such that certain powers of elements of $M_n(F)$ are elements of $T$, and only they, where $M_n(F)$ is the ideal of identities of the algebra $F_n$ of all $(n\times n)$-matrices over $F$. In this case a $T$-ideal can also be defined as a (two-sided) ideal that is closed under all differentiations of the free algebra. The quotient algebra $F[X]/T$ is a [[PI-algebra|PI-algebra]] with $T$ as set of polynomial identities. It is called the relatively free algebra (or generic algebra) with $T$-ideal of identities $T$ (and is a free algebra in the variety of algebras defined by the identities in $T$). The algebra $F[X]/T$ has no zero divisors if and only if $T=M_n(F)$ for some natural number $n$. Every $T$-ideal $T$ of a free associative algebra is primary.
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202032.png" />-ideals of a free associative algebra on infinitely many generators over a field of characteristic zero form a free semi-group under the operation of multiplication of ideals. In this case a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202033.png" />-ideal can be defined as an ideal invariant under all automorphisms of the free algebra.
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The $T$-ideals of a free associative algebra on infinitely many generators over a field of characteristic zero form a free semi-group under the operation of multiplication of ideals. In this case a $T$-ideal can be defined as an ideal invariant under all automorphisms of the free algebra.
  
For the question as to whether every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202034.png" />-ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202035.png" /> is the totally invariant closure of finitely many elements (Specht's problem) see also [[Variety of rings|Variety of rings]].
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For the question as to whether every $T$-ideal of $F[X]$ is the totally invariant closure of finitely many elements (Specht's problem) see also [[Variety of rings|Variety of rings]].
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202036.png" />-ideals can be defined for non-associative algebras (Lie, alternative and others) by analogy with the associative case.
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$T$-ideals can be defined for non-associative algebras (Lie, alternative and others) by analogy with the associative case.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
A.R. Kemer has positively solved the Specht problem in the case of characteristic zero (see [[Variety of rings|Variety of rings]]). He has also introduced the notion of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202039.png" />-prime ideal, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202040.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202041.png" />) for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202042.png" />-ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202043.png" />, with different variables in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202045.png" />, then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202046.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202047.png" />. Similarly, for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202049.png" />-nilpotent ideal. He has shown that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202050.png" />-ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202051.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202052.png" />-ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202053.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202054.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202055.png" />-nilpotent and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202056.png" /> is a finite product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202057.png" />-prime ideals.
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A.R. Kemer has positively solved the Specht problem in the case of characteristic zero (see [[Variety of rings|Variety of rings]]). He has also introduced the notion of a $T$-prime ideal, i.e. if $f[x_1,\dots,x_n]g[x_{n+1},\dots,x_m]\equiv0$ ($\bmod\,P$) for a $T$-ideal $P$, with different variables in $f$ and $g$, then either $f[x_1,\dots,x_n]\in P$ or $g[x_{n+1},\dots,x_m]\in P$. Similarly, for a $T$-nilpotent ideal. He has shown that for every $T$-ideal $I$ there exists a $T$-ideal $N(I)\supset I$ such that $N(I)/I$ is $T$-nilpotent and $N(I)$ is a finite product of $T$-prime ideals.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.R. Kemer,  "Solution of the finite basis problem"  ''Soviet Math. Dokl.'' , '''37'''  (1988)  pp. 60–64  ''Dokl. Akad. Nauk SSSR'' , '''298'''  (1988)  pp. 273–277</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.R. Kemer,  "Finite basis property of identities of associative algebras"  ''Algebra and Logic'' , '''26'''  (1987)  pp. 362–397  ''Algebra i Logika'' , '''26'''  (1987)  pp. 597–641</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Formanek,  "The polynomial identities and invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202058.png" /> matrices" , Amer. Math. Soc.  (1991)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.R. Kemer,  "Solution of the finite basis problem"  ''Soviet Math. Dokl.'' , '''37'''  (1988)  pp. 60–64  ''Dokl. Akad. Nauk SSSR'' , '''298'''  (1988)  pp. 273–277</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.R. Kemer,  "Finite basis property of identities of associative algebras"  ''Algebra and Logic'' , '''26'''  (1987)  pp. 362–397  ''Algebra i Logika'' , '''26'''  (1987)  pp. 597–641</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Formanek,  "The polynomial identities and invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092020/t09202058.png" /> matrices" , Amer. Math. Soc.  (1991)</TD></TR></table>

Latest revision as of 08:33, 29 August 2014

of a free associative algebra

A totally invariant ideal, that is, an ideal invariant under all endomorphisms. The set of all polynomial identities of an arbitrary variety of associative algebras over a field $F$ (cf. Associative rings and algebras) forms a $T$-ideal in the countably-generated free algebra $F[X]$, $X=\{x_1,\dots,x_k,\dots\}$. Thus, there exists a one-to-one correspondence between the $T$-ideals of $F[X]$ and the varieties of associative algebras over $F$. If $F$ has characteristic $0$, then for every $T$-ideal $T\subseteq F[X]$ there exists a natural number $n=n(T)$ such that certain powers of elements of $M_n(F)$ are elements of $T$, and only they, where $M_n(F)$ is the ideal of identities of the algebra $F_n$ of all $(n\times n)$-matrices over $F$. In this case a $T$-ideal can also be defined as a (two-sided) ideal that is closed under all differentiations of the free algebra. The quotient algebra $F[X]/T$ is a PI-algebra with $T$ as set of polynomial identities. It is called the relatively free algebra (or generic algebra) with $T$-ideal of identities $T$ (and is a free algebra in the variety of algebras defined by the identities in $T$). The algebra $F[X]/T$ has no zero divisors if and only if $T=M_n(F)$ for some natural number $n$. Every $T$-ideal $T$ of a free associative algebra is primary.

The $T$-ideals of a free associative algebra on infinitely many generators over a field of characteristic zero form a free semi-group under the operation of multiplication of ideals. In this case a $T$-ideal can be defined as an ideal invariant under all automorphisms of the free algebra.

For the question as to whether every $T$-ideal of $F[X]$ is the totally invariant closure of finitely many elements (Specht's problem) see also Variety of rings.

$T$-ideals can be defined for non-associative algebras (Lie, alternative and others) by analogy with the associative case.

References

[1] C. Procesi, "Rings with polynomial identities" , M. Dekker (1973)
[2] N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
[3] I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968)
[4] S. Amitsur, "The -ideals of the free ring" J. London Math. Soc. , 30 (1955) pp. 470–475
[5] W. Specht, "Gesetze in Ringen I" Math. Z. , 52 (1950) pp. 557–589
[6] G. Bergman, J. Lewin, "The semigroup of ideals of a fir is (usually) free" J. London Math. Soc. (2) , 11 : 1 (1975) pp. 21–31


Comments

A.R. Kemer has positively solved the Specht problem in the case of characteristic zero (see Variety of rings). He has also introduced the notion of a $T$-prime ideal, i.e. if $f[x_1,\dots,x_n]g[x_{n+1},\dots,x_m]\equiv0$ ($\bmod\,P$) for a $T$-ideal $P$, with different variables in $f$ and $g$, then either $f[x_1,\dots,x_n]\in P$ or $g[x_{n+1},\dots,x_m]\in P$. Similarly, for a $T$-nilpotent ideal. He has shown that for every $T$-ideal $I$ there exists a $T$-ideal $N(I)\supset I$ such that $N(I)/I$ is $T$-nilpotent and $N(I)$ is a finite product of $T$-prime ideals.

References

[a1] A.R. Kemer, "Solution of the finite basis problem" Soviet Math. Dokl. , 37 (1988) pp. 60–64 Dokl. Akad. Nauk SSSR , 298 (1988) pp. 273–277
[a2] A.R. Kemer, "Finite basis property of identities of associative algebras" Algebra and Logic , 26 (1987) pp. 362–397 Algebra i Logika , 26 (1987) pp. 597–641
[a3] E. Formanek, "The polynomial identities and invariants of matrices" , Amer. Math. Soc. (1991)
How to Cite This Entry:
T-ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=T-ideal&oldid=14127
This article was adapted from an original article by V.N. Latyshev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article