Illinois Journal of Mathematics

Homomorphic images of pro-nilpotent algebras

George M. Bergman

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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.

Article information

Illinois J. Math., Volume 55, Number 3 (2011), 719-748.

First available in Project Euclid: 29 May 2013

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Zentralblatt MATH identifier

Primary: 16N20: Jacobson radical, quasimultiplication 16N40: Nil and nilpotent radicals, sets, ideals, rings 17A01: General theory 18A30: Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
Secondary: 13C13: Other special types 17B30: Solvable, nilpotent (super)algebras


Bergman, George M. Homomorphic images of pro-nilpotent algebras. Illinois J. Math. 55 (2011), no. 3, 719--748. doi:10.1215/ijm/1369841782.

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