P.191, end of next-to-last paragraph: after reference "" 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 "": "*R. Freese  has proved that the free modular lattice on 5 generators has unsolvable word problem."
P.213, first two lines: End these with a ")" after "". (Reference  has been changed; the comments applied to the earlier reference. The new reference is given under "Further items" below.)
P.215, reference : change "in preparation" to "unpublished".
P.217, reference : change "to appear" to "Annals of Math. Logic 17 (1979) 117-150".
P.218. The conference proceedings referred to in reference  appeared under the title Word Problems; the article in question occurs in vol.II, pp.87-100 thereof, 1980.
Add reference : "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.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  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 , 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.216, reference : Change "1971" to "1st edition, 1971, 2nd edition 1985".
P.217, reference : "M. W. Milnor" should be "J. W. Milnor".
P.218: Change reference  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  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  change "to appear" to "unpublished".
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