### Updates to

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

[2] Linear maps on *k*^{I},
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 [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 *k*^{N} 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 [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*)^{ℵ0}".

### Errata

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
**Z**^{N} → Z factors through finitely
many coordinates.
That paper studied, rather subgroups of **Z**^{N}.
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.
DOI.
arXiv:1301.6383.
abstract.
MR3226022

[4] George M. Bergman,
*Homomorphisms on**
infinite direct products
of groups, rings
and monoids,*
Pacific Journal of Mathematics, **274** (2015) 451-495.
DOI.
arXiv:1406.1932.
abstract.
MR3332912.

[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
k*^{N}* of maximal dimension*,
preprint, 2016, 2 pp.
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