Updates to

[1] Homomorphisms on infinite direct product algebras, especially Lie algebrasJ. Alg. 333 (2011) 67-104, and

[2] Linear maps on  kI,  and homomorphic images of infinite direct product algebrasJ. Alg. 356 (2012) 257-274,

by George M. Bergman and Nazih Nahlus

The MathSciNet review of [1] observes that a positive answer to the question posed in the 4th paragraph of §13.8, whether every finite-dimensional Lie algebra over a field of positive characteristic is a homomorphic image of a finite-dimensional semisimple Lie algebra, can be obtained from Block's theorem (§3.3 of H.Strade, Simple Lie Algebras over Fields of Positive Characteristic, I). 

In [3] below, I show that the results of [1] and [2] above can be used to get stronger results:  that where we prove factorization through finitely many factors under the assumption that  I  has cardinality less than a cardinal  κ,  the same can be deduced under the weaker assumption that  I  has cardinality less than all measurable cardinals  ≥ κ.  In particular, applying this to a result in [2] yields an affirmative answer to [1, Question 19]. 

Fares Maalouf [5] abstracts the property, obtained in [1] and [2] for various classes of algebras  B,  that whenever one has a homomorphism from a countable direct product of algebras into  B,  the induced homomorphism to  B / Z(B)  factors through a finite subproduct, calling such algebras  B  weakly slender, and obtains further properties of such algebras.  He also proves a strengthening [5, first part of proof of Theorem 7] of the final statement of [2, Lemma 6], namely, he shows that if  k  is an infinite field, then any k-linear map from  kN  to a vector space of dimension  < card(k)0  has in its kernel an element with cofinite support.  This leads to an affirmative answer to the question raised at the end of the 3rd paragraph of p.273 of [2]. 

In [4], I investigate analogous questions concerning homomorphisms on direct products of groups, abelian groups, rings, and monoids. 

In [6], Maalouf partly answers the question raised in the third paragraph of section 7 of [2]; namely, he shows that in the case where the set  I  is countable, we can indeed weaken the hypothesis referred to there all the way to "< card(k)0". 


In [1], p.89, first sentence of 3rd paragraph of section 11.5, I was wrong in referring to Specker's paper there called [34] for the result that a homomorphism  ZN → Z  factors through finitely many coordinates.  That paper studied, rather subgroups of  ZN.  The study of slender groups, of which that result is the simplest instance, seems to be due to Łoś. 

In [1], p.96, last sentence of section 13.1, "of nilpotent" should be "of nilpotent algebras". 

In [2], 3rd paragraph of section 7, the reference for the Erdős-Kaplansky theorem should show page-number 247, not 246. 

Additional references

[3] George M. Bergman, Families of ultrafilters, and homomorphisms on infinite direct product algebras, J. Symb. Log. 79 (2014) 223–239.  DOIarXiv:1301.6383abstractMR3226022
[4] George M. Bergman, Homomorphisms on infinite direct products of groups, rings and monoids, Pacific Journal of Mathematics, 274 (2015) 451-495.  DOIarXiv:1406.1932abstractMR3332912
[5] Fares Maalouf, Weakly slender algebras, Journal of Pure and Applied Algebra, 218 (2014) 1754-1759.  MR3188870
[6] Fares Maalouf, A pairwise almost disjoint subspace of kN of maximal dimension, preprint, 2016, 2 pp. 

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