Illinois Journal of Mathematics

Homomorphic images of pro-nilpotent algebras

George M. Bergman

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 29 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1369841782

Digital Object Identifier
doi:10.1215/ijm/1369841782

Mathematical Reviews number (MathSciNet)
MR3069281

Zentralblatt MATH identifier
1285.17001

Subjects
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

Citation

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


Export citation

References

  • H. Bass, Algebraic K-theory, W. A. Benjamin, New York–Amsterdam, 1968.
  • G. M. Bergman, An invitation to general algebra and universal constructions, Henry Helson, Berkeley, CA, 1998. Available at http://math.berkeley.edu/~gbergman/245.
  • G. M. Bergman, Continuity of homomorphisms on pro-nilpotent algebras, Illinois J. Math. 55 (2011), 749–770.
  • G. M. Bergman and A. O. Hausknecht, Cogroups and co-rings in categories of associative rings, Mathematical Surveys and Monographs, vol. 45, Amer. Math. Soc., Providence, RI, 1996.
  • G. M. Bergman and N. Nahlus, Homomorphisms on infinite direct product algebras, especially Lie algebras, J. Algebra 333 (2011), 67–104. Available at http://math. berkeley.edu/~gbergman/papers.
  • G. M. Bergman and N. Nahlus, Linear maps on $k^I$, and homomorphic images of infinite direct product algebras, J. Algebra 356 (2012), 257–274. Available at http:// math.berkeley.edu/~gbergman/papers.
  • K. Bowman and D. A. Towers, On almost nilpotent-by-abelian Lie algebras, Linear Algebra Appl. 247 (1996), 159–167.
  • A. L. S. Corner, Three examples of hopficity in torsion-free abelian groups, Acta Math. Acad. Sci. Hungar. 16 (1965), 303–310.
  • J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic pro-$ p $ groups, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 61, Cambridge University Press, Cambridge, 1999.
  • L. Henkin, A problem on inverse mapping systems, Proc. Amer. Math. Soc. 1 (1950), 224–225.
  • G. Higman and A. H. Stone, On inverse systems with trivial limits, J. London Math. Soc. 29 (1954), 233–236.
  • V. A. Hiremath, Hopfian rings and Hopfian modules, Indian J. Pure Appl. Math. 17 (1986), 895–900. Available at http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_2/20005a4f_895.pdf.
  • K. H. Hofmann and S. A. Morris, The Lie theory of connected pro-Lie groups. A structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups, EMS Tracts in Mathematics, vol. 2, European Mathematical Society, Zürich, 2007.
  • N. Jacobson, Lie algebras, Dover, New York, 1979.
  • T. Y. Lam, A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131, Springer, New York, 1991.
  • T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer, New York, 1999.
  • S. Lang, Algebra, 3rd ed., Addison-Wesley, Reading, 1993; reprinted as Graduate Texts in Mathematics, vol. 211, Springer, New York, 2002.
  • S. Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5, Springer, New York, 1971.
  • E. Sąsiada and P. M. Cohn, An example of a simple radical ring, J. Algebra 5 (1967), 373–377.
  • R. D. Schafer, An introduction to nonassociative algebras, Dover, New York, 1995.
  • J.-P. Serre, Galois cohomology, Springer, Berlin, 1997.
  • A. M. Slin'ko, The equivalence of certain nilpotencies of right-alternative rings, Algebra Logika 9 (1970), 342–348 (in Russian).
  • K. Varadarajan, Hopfian and co-Hopfian objects, Publ. Mat. 36 (1992), 293–317.
  • W. C. Waterhouse, An empty inverse limit, Proc. Amer. Math. Soc. 36 (1972) 618.