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 e12, y 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 X∈Mn(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.
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 .