PI-subrings and algebraic elements in enveloping algebras and their fields of fractions. (English) Zbl 0672.17009
Let R be a (not necessarily commutative) subfield of the field D constructed by P. M. Cohn to embed the universal enveloping algebra U(L) of an arbitrary Lie algebra L over a commutative field F. Suppose that Z is the centre of R and E, \(R\supset E\supset Z\), is a commutative subfield, finite dimensional over Z. The author shows that if char F\(=0\) then \(E=Z\) and if char F\(=p>0\) then \(\dim_ Z E\) is a power of p. The main tool in the proof of this result is the valuation theory of D started by P. M. Cohn and essentially refined by the author.
As corollaries, there are some results on PI-subrings of D and of U(L), on algebraic elements in subrings of D and some others. For instance, a PI-subring S of U(L) is commutative if char F\(=0\) and its PI-degree is a power of p if char F\(=p>0\).
As corollaries, there are some results on PI-subrings of D and of U(L), on algebraic elements in subrings of D and some others. For instance, a PI-subring S of U(L) is commutative if char F\(=0\) and its PI-degree is a power of p if char F\(=p>0\).
Reviewer: Yu.Bakhturin
Keywords:
field of fractions; noncommutative valuations; universal enveloping algebra; PI-subrings; algebraic elementsReferences:
| [1] | Jacobson, N., Lie Algebras (1962), Wiley: Wiley New York/London · JFM 61.1044.02 |
| [2] | Cohn, P. M., On the embedding of rings in skew fields, (Proc. London Math. Soc., 11 (1961)), 511-530 · Zbl 0104.03203 |
| [3] | Lichtman, A. I., The residual nilpotence of multiplicative groups of skew fields generated by universal enveloping algebras, J. Algebra, 112, 250-263 (1988) · Zbl 0636.16007 |
| [4] | Posner, E., Prime rings satisfying a polynomial identity, (Proc. Amer. Math. Soc., 11 (1960)), 180-184 · Zbl 0215.38101 |
| [5] | Rowen, S. L., Polynomial Identities in Ring Theory (1980), Academic Press: Academic Press Orlando, FL · Zbl 0461.16001 |
| [6] | Bahturin, Yu. A., Identities in Lie Algebras (1985), Nauka: Nauka Moscow, (in Russian) · Zbl 0571.17001 |
| [7] | Bahturin, Yu. A., Identities in the universal envelopes of Lie algebras, J. Austral. Math. Soc., 18, 19-27 (1974) |
| [8] | Lichtman, A. I., Matrix rings and linear groups over a field of fractions of enveloping algebras and group rings, I, J. Algebra, 88, 1-37 (1984) · Zbl 0536.16023 |
| [9] | Lichtman, A. I., Growth in enveloping algebras, Israel J. Math., 47, 296-304 (1984) · Zbl 0552.17006 |
| [10] | Borevich, Z. I.; Shafarevich, I. R., Number Theory, (Pure and Applied Mathematics, Vol. 20 (1966), Academic Press: Academic Press New York) · Zbl 0121.04202 |
| [11] | Zarisski, O.; Samuel, P., (Commutative Algebra, Vol. 2 (1960), Van Nostrand: Van Nostrand Princeton, NJ) · Zbl 0121.27801 |
| [12] | Jacobson, N., Structure of rings, (AMS Colloq. Pub., Vol. 37 (1956), Amer. Math. Soc: Amer. Math. Soc Providence, RI) · JFM 65.1131.01 |
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