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  md2  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  e12y e21,  and  x e22

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  XMn(R′) which is not invertible in  Mn(R′),  but is so in  Mn(R*).  Namely,  R′,  n  and  X  are as described in this section, while for  R*  we may take the overring of all Md(F)-valued functions on  S

P.267, line 6:  "Mn(F)-valued" should be "Md(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 .