Homomorphisms on infinite direct product algebras, especially Lie algebras, J. Alg. 333 (2011) 67-104, and
 Linear maps on kI, and homomorphic images of infinite direct product algebras, J. Alg. 356 (2012) 257-274,
by George M. Bergman and Nazih Nahlus The MathSciNet review of  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  below, I show that the results of  and  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  yields an affirmative answer to [1, Question 19].
Fares Maalouf  abstracts the property, obtained in  and  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)ℵ0 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 .
In , I investigate analogous questions concerning homomorphisms on direct products of groups, abelian groups, rings, and monoids.
In , Maalouf partly answers the question raised in the third paragraph of section 7 of ; 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)ℵ0".
In , p.96, last sentence of section 13.1, "of nilpotent" should be "of nilpotent algebras".
In , 3rd paragraph of section 7, the reference for the Erdős-Kaplansky theorem should show page-number 247, not 246.
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