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Fall 2011 Homomorphic images of pro-nilpotent algebras
George M. Bergman
Illinois J. Math. 55(3): 719-748 (Fall 2011). DOI: 10.1215/ijm/1369841782


It is shown that any finite-dimensional homomorphic image of an inverse limit of nilpotent not-necessarily-associative algebras over a field is nilpotent. More generally, this is true of algebras over a general commutative ring $k$, with “finite- dimensional” replaced by “of finite length as a $k$-module.”

These results are obtained by considering the multiplication algebra $M(A)$ of an algebra $A$ (the associative algebra of $k$-linear maps $A → A$ generated by left and right multiplications by elements of $A$), and its behavior with respect to nilpotence, inverse limits, and homomorphic images.

As a corollary, it is shown that a finite-dimensional homomorphic image of an inverse limit of finite-dimensional solvable Lie algebras over a field of characteristic 0 is solvable.

It is also shown by example that infinite-dimensional homomorphic images of pro-nilpotent algebras can have properties far from those of nilpotent algebras; in particular, properties that imply that they are not residually nilpotent.

Several open questions and directions for further investigation are noted.


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George M. Bergman. "Homomorphic images of pro-nilpotent algebras." Illinois J. Math. 55 (3) 719 - 748, Fall 2011.


Published: Fall 2011
First available in Project Euclid: 29 May 2013

zbMATH: 1285.17001
MathSciNet: MR3069281
Digital Object Identifier: 10.1215/ijm/1369841782

Primary: 16N20, ‎16N40, 17A01, 18A30
Secondary: 13C13, 17B30

Rights: Copyright © 2011 University of Illinois at Urbana-Champaign


Vol.55 • No. 3 • Fall 2011
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