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P.215, reference [10]: change "in preparation" to "unpublished".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 *S*_{1} is compatible" to "such that
that *S*_{2} 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 f*_{σ} *B*
should be *A f*_{σ} *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:
*r*_{σ(a)} should be
*f*_{σ(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 *R _{h}*
with weak algorithm as in [20, section 2.6], one can get a reduction
system for an ordinary

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".