##### Errata and updates to

### George M. Bergman, *Some examples in PI ring theory*,
Israel J. Math. **18** (1974) 257-277

### Errata, addenda, and clarifications

P.261, Question 5: Lance Small has proved
that the answer is affirmative if *A* is Noetherian.

P.261, paragraph following Question 5: For "has found
an example of", it would be better to say "has shown the
existence of", since the argument (on p.7 of [9]) is based
on displaying uncountably many rings *R* / *I*, of
which only countably many are
embeddable in matrix rings over commutative **Q**-algebras.
To see that countability, note that if *S*
is an *m*-generated **Q**-algebra
of *d*×*d* matrices
over a commutative algebra *A*, then *A* can be taken
to be generated by *md*^{2} elements,
and there are only countably many possibilities for the
finite number of relations presenting *A* in terms of
these generators.
To see that the ring *R* described is affine, one verifies
that it is generated by *e*_{12},
*y e*_{21}, and *x e*_{22}.

Pp.266-270, construction **C.**: Warren Dicks has pointed out to
me that this gives an example
of a ring *R*′ which is *division closed* in
an over-ring *R**, meaning that whenever an
element of *R*′ is invertible
in *R**, its inverse already
lies in *R*′, but which is not
*rationally closed* therein, meaning that
for some *n* there is
a matrix *X*∈*M*_{n}(*R*′)
which is not invertible in *M*_{n}(*R*′), but
is so in *M*_{n}(*R**).
Namely, *R*′, *n* and *X* are as described
in this section, while for *R** we may take the overring of
all *M*_{d}(*F*)-valued functions on *S*.

P.267, line 6: "*M*_{n}(*F*)-valued"
should be "*M*_{d}(*F*)-valued".

P.267, 2nd paragraph, line 6: "shoold" should be "should".

P.267, 2nd paragraph, first word of last line: for "those"
read "the regular elements".

P.268, 2nd line after top display: To see the assertion
introduced by "Hence", remember that *P* is a polynomial
in one variable.

### Updates to three references

4.
G. M. Bergman,
*Rational relations and rational identities in division algebras,
I*, J. Alg. **43** (1976), 252-266.
DOI.
MR 55#56853a .

5.
G. M. Bergman and Lance W. Small,
*P.i. degrees and prime ideals*,
J. Alg. **33** (1975) 435-462.
DOI.
MR 50#
13129 .

9.
L. W. Small,
*Rings satisfying a polynomial identity.*
Written from notes taken by Christine Bessenrodt.
Vorlesungen aus dem Fachbereich Mathematik der Universität Essen, 5.
Universität Essen, Fachbereich Mathematik, Essen, 1980. 38 pp.
MR 82j#
16028 .

DOI.
Back to publications-list.
MR 50#9956 .