George M. Bergman, The diamond lemma for ring theory, Advances in Mathematics 29 (1978) 178-218.


There has been a considerable amount of related work, initiated independently by Buchberger et al., under the name Groebner bases, which I have not followed.  This mostly, but not exclusively, concerns commutative algebras.  E.g., see T. Becker and V. Weispfenning, Gröbner Bases.  A computational approach to commutative algebra, Graduate Texts in Mathematics, v.141, Springer-Verlag, 1993, MR 95e:13018. 

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

Corrections already made in version duplicated and sent out in '80's:

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P.215, reference [10]: change "in preparation" to "unpublished".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". 

Erroneous correction made in version duplicated in 1993:

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. 

Further corrections and updates:

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.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". 

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