A matrix representation for associative algebras. I

Lewin, Jacques

doi:10.1090/S0002-9947-1974-0338081-5

A matrix representation for associative algebras. I
HTML articles powered by AMS MathViewer

by Jacques Lewin

Trans. Amer. Math. Soc. 188 (1974), 293-308

DOI: https://doi.org/10.1090/S0002-9947-1974-0338081-5

PDF | Request permission

Abstract:

Let F be a mixed free algebra on a set X over the field K. Let U, V be two ideals of F, and $\{ \delta (x),(x \in X)\}$ a basis for a free $(F/U,F/V)$-bimodule T. Then the map $x \to (\begin {array}{*{20}{c}} {x + V} & 0 \\ {\delta (x)} & {x + U} \\ \end {array} )$ induces an injective homomorphism $F/UV \to (\begin {array}{*{20}{c}} {F/V} & 0 \\ T & {F/U} \\ \end {array} )$. If $F/U$ and $F/V$ are embeddable in matrices over a commutative algebra, so is $F/UV$. Some special cases are investigated and it is shown that a PI algebra with nilpotent radical satisfies all identities of some full matrix algebra.

References

S. A. Amitsur, The identities of $\textrm {PI}$-rings, Proc. Amer. Math. Soc. 4 (1953), 27–34. MR 52397, DOI 10.1090/S0002-9939-1953-0052397-3
S. A. Amitsur, A noncommutative Hilbert basis theorem and subrings of matrices, Trans. Amer. Math. Soc. 149 (1970), 133–142. MR 258869, DOI 10.1090/S0002-9947-1970-0258869-5
George M. Bergman and Warren Dicks, On universal derivations, J. Algebra 36 (1975), no. 2, 193–211. MR 387353, DOI 10.1016/0021-8693(75)90098-8
D. E. Cohen, On the laws of a metabelian variety, J. Algebra 5 (1967), 267–273. MR 206104, DOI 10.1016/0021-8693(67)90039-7
P. M. Cohn, On a generalization of the Euclidean algorithm, Proc. Cambridge Philos. Soc. 57 (1961), 18–30. MR 118743, DOI 10.1017/s0305004100034812
Paul M. Cohn, On a class of rings with inverse weak algorithm, Math. Z. 117 (1970), 1–6. MR 279123, DOI 10.1007/BF01109821
Ralph H. Fox, Free differential calculus. I. Derivation in the free group ring, Ann. of Math. (2) 57 (1953), 547–560. MR 53938, DOI 10.2307/1969736
Karl W. Gruenberg, Cohomological topics in group theory, Lecture Notes in Mathematics, Vol. 143, Springer-Verlag, Berlin-New York, 1970. MR 0279200, DOI 10.1007/BFb0059162
C. K. Gupta and N. D. Gupta, On the linearity of free nilpotent-by-abelian groups, J. Algebra 24 (1973), 293–302. MR 310061, DOI 10.1016/0021-8693(73)90139-7
Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
John P. Labute, Algèbres de Lie et pro-$p$-groupes définis par une seule relation, Invent. Math. 4 (1967), 142–158 (French). MR 218495, DOI 10.1007/BF01425247
V. N. Latyšev, Generalization of Hilbert’s theorem on the finiteness of bases, Sibirsk. Mat. Ž. 7 (1966), 1422–1424 (Russian). MR 0202746
Jacques Lewin, On some infinitely presented associative algebras, J. Austral. Math. Soc. 16 (1973), 290–293. Collection of articles dedicated to the memory of Hanna Neumann, III. MR 0340329, DOI 10.1017/S1446788700015068
I. V. L′vov, Maximality conditions in algebras with identity relations, Algebra i Logika 8 (1969), 449–459 (Russian). MR 0279127
Roger C. Lyndon, Cohomology theory of groups with a single defining relation, Ann. of Math. (2) 52 (1950), 650–665. MR 47046, DOI 10.2307/1969440
Wilhelm Magnus, On a theorem of Marshall Hall, Ann. of Math. (2) 40 (1939), 764–768. MR 262, DOI 10.2307/1968892
Lance W. Small, An example in $\textrm {PI}$-rings, J. Algebra 17 (1971), 434–436. MR 272823, DOI 10.1016/0021-8693(71)90025-1