Two questions have been answered, Question 6 and Question 18:

**Solution to Question 6, p.40.**
For *R* a ring and
(*M _{i}*)

The idea will be to take a family of rings
(*R _{i}*)

The discussion following the question shows that the difficulty is
somehow located in the vicinity of finitely many nonprincipal
ultrafilters on the index-set ω, without loss of
generality, just one; so let us push the problem back one more
level: Let us try to find a family of rings
(*R _{i}*)

To find such *S´* and *S*, we
use the fact that it is easy to get such an example for vector
spaces, since any infinite-dimensional vector space has a subspace
of countably infinite codimension.
So let *k* be any field,
let *V* be an infinite-dimensional vector
space over *k,* make the direct
sum *k* ⊕ *V* into a
commutative ring in the standard way (using the multiplication
of *k,* the scalar multiplication relating
*k* and *V,* and the zero multiplication
on *V* ), and use this ring for each of
the *R _{i}.*
The ultraproduct of these rings with respect to any nonprincipal
ultrafilter on ω will be a ring of the
form

If we would like our ring *R* to be an integral
domain, we can achieve this by replacing
the *R* obtained above by any integral domain
that can be mapped surjectively to it; e.g., a polynomial
ring in sufficiently many indeterminates.
We can make our module-structures "nicer" in the same way,
replacing, in each of the ring-and-module pairs
*R _{i} *,

Something it is not clear is whether we can do is indicated in part (1) of the following question. An affirmative answer to part (2) would imply an affirmative answer to part (1).

**Question 6´.** (1)* If R is a ring and
*(*M _{i}*)

**Solution to Question 18, p.47.**
This was answered by Adam Megacz.
The question concerns an algebra *M* all of
whose operations have arities < λ
for a regular infinite cardinal λ.
Theorem 16 shows that for any κ ≥ λ,
*M*^{κ} requires
either > κ or < λ generators;
and Question 18 asks whether in this
statement < λ
can be replaced by < ℵ_{0}.
Megacz shows that when λ is the
successor μ^{+} of a
cardinal μ < κ,
the bound of that theorem cannot be improved at all.
The following somewhat
simplifies and generalizes his construction.

Let κ^{κ} denote the set of all
functions κ → κ, and for each
*f* ∈ κ^{κ},
let α_{f} be the μ-ary operation
on κ such that for any μ-tuple *x* =
(*x _{i }*)

α_{f }(*x*) =
*x*_{0}
if *x* has < μ distinct components,

α_{f }(*x*) =
*f* (*x*_{0})
if *x* has μ distinct components.

Let *A* be the algebra with underlying
set κ and the above
operations α_{f} ,
which all have arities μ < λ.
Applied to any subset of cardinality < μ, these
operations give nothing new, so *A* cannot be generated
by < μ elements, hence neither
can *A*^{κ}.
However, *A*^{κ} can be
generated by μ elements, namely the identity
map κ → κ, and any μ
distinct constants (or indeed, any family
of μ elements which together achieve μ distinct
values at each coordinate *i* ∈κ).
For, given an element *f* ∈ *A*^{κ} =
κ^{κ} that
one wants to obtain, one takes the corresponding operation
α_{f} and applies it
to the μ-tuple having first entry the identity map, and
remaining entries the μ other elements described above.

Hence, though *A* can be generated
by ≤ κ elements, and therefore,
as shown in Theorem 16,
by < λ elements, namely, by μ elements,
it cannot be generated by a set of any smaller cardinality.

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