A result essentially equivalent to the Theorem 1.2 of my paper had appeared (unknown to me) two years earlier as Proposition 1 of L. A. Bokut', Imbeddings into simple associative algebras (Russian), Algebra i Logika 15 (1976) 117--142, 245, translated in Algebra and Logic 15 (1976) 73--90, MR58#22167. Bokut' describes this as a result of Shirshov, who had published a version for Lie algebras in A. I. Shirshov, Some algorithmic problems for Lie algebras, Sibirski. Math. Zh., 2 (1962) 291--296, MR0183753, English translation in SIGSAM Bull. 33(2) (1999) 3–6. Bokut' has subsequently called this technique ``the method of Gröbner-Shirshov bases'', e.g., in L. A. Bokut, Yuqun Chen and Qiuhui Mo, Gröbner-Shirshov bases and embeddings of algebras, Internat. J. Algebra Comput. 20 (2010) 875-900, MR2738549.
P.191, end of next-to-last paragraph: after reference "[75]" add "section 3".
P.209. On 6th line of section 10.5, change "by matrices" to "by r n×n matrices". On 3rd line from bottom, both occurrences of boldface L should be fraktur L (as on p.186).
P.210, line 7: add footnote to "[61]": "*R. Freese [79] has proved that the free modular lattice on 5 generators has unsolvable word problem."
P.213, first two lines: End these with a ")" after "[74]". (Reference [74] has been changed; the comments applied to the earlier reference. The new reference is given under "Further items" below.)
P.215, reference [10]: change "in preparation" to "unpublished".
P.217, reference [46]: change "to appear" to "Annals of Math. Logic 17 (1979) 117-150".
P.218. The conference proceedings referred to in reference [67] appeared under the title Word Problems; the article in question occurs in vol.II, pp.87-100 thereof, 1980.
Add reference [79]: "R. Freese, Free modular lattices, Trans.A.M.S. 261 (1980) 81-91".
P.183, fifth line of Corollary 1.3: I incorrectly changed "such that S1 is compatible" to "such that that S2 is compatible". Sorry - it was right to begin with.
P.179: Condition (ii) is just 3 lines long; so there should be a space after the third line of that condition.
P.180, 2nd line of 3rd paragraph of section 1: Where I have "We shall call a irreducible", the "a" should be italic; it is an element of k<X>, correctly shown in italics both on the preceding line and earlier on this line.
P.181, two lines before Theorem 1.2: A fσ B should be A fσ C.
P.181, third paragraph (proof of (ii)): To reduce by induction to the case where r is a single reduction, one uses an inductive statement slightly stronger than (ii), namely that the product a r(b)c has the same property assumed for the product abc .
P.187, last sentence of section 3: The question mentioned here, of whether two Lie algebras over a field which have isomorphic universal enveloping algebras are themselves isomorphic, has been answered in the negative; see David Riley and Hamid Usefi, The isomorphism problem for universal enveloping algebras of Lie algebras, Algebr. Represent. Theory, 10 (2007) 517--532. MR 2008i:17015. In the example given there, one of the two Lie algebras is in fact free. On the other hand, the base field there is of positive characteristic; so far as I know, the question is open in characteristic 0.
P.189, statement of Corollary 4.2: In the first display, (zy, yz + a) should be (zy, yz − a), and similarly, (yx, xy + b) should be (yx, xy − b).
P.196, third line of Theorem 6.1: rσ(a) should be fσ(a).
P.198, line 5: "Theorem 7.1" should be "Theorem 6.1".
P.205: In section 9.7, the references to [20] refer to the first edition. In the second edition, the corresponding material may be found in section 2.6, and the Exercise referred to toward the end becomes Exercise 4 on p.114. (Further adjustments may be needed when the third edition appears.)
However, to get the result needed for [20], the adaptation of the Diamond Lemma to truncated filtered rings developed in section 9.7 is not really necessary. Rather, from a truncated filtered ring Rh with weak algorithm as in [20, section 2.6], one can get a reduction system for an ordinary k-ring of the sort dealt with in Proposition 7.1 of this paper. The set of elements of this k-ring of formal degree ≤ h+1 then give the desired height-h+1 truncated filtered ring Rh+1 . (This construction would not give the universal height-h+1 extension of an arbitrary height-h truncated filtered ring, but as noted in the last paragraph of section 9.7, the weak algorithm leads to reduction systems of a particularly simple sort.)
P.215, reference [8]: This has appeared, greatly extended, as
An Invitation to General Algebra and Universal Constructions,
2015, Springer Universitext, 572 pp.
DOI 10.1007/978-3-319-11478-1,
ISBN9 78-3-319-11477-4,
eBook ISBN 978-3-319-11478-1.
(First edition: ISBN 0-9655211-4-1, published by Henry Helson, 1998
MR 99h:18001 .)
However, to my embarrassment, I haven't been able to find anywhere
in this paper where [8] is referred to, even
pattern-searching a digitized copy.
P.216, reference [20]: Change "1971" to "1st edition, 1971, 2nd edition 1985".
P.217, reference [42]: "M. W. Milnor" should be "J. W. Milnor".
P.218: Change reference [74] to: "Warren Dicks and I.J.Leary, Exact sequences for mixed coproduct/tensor product ring constructions, Publ. Sec. Math. Univ. Autònoma Barcelona 38 (1994) 89-126." Change reference [75] to: "George M. Bergman and Samuel M. Vovsi, Embedding rings in completed graded rings, 2. Algebras over a field, J. Alg. 84 (1983) 25-41". In reference [76] change "to appear" to "unpublished".