MAJOR ERRATA IN THE ORIGINAL TWO-VOLUME EDITION “RING THEORY”

VOLUME I.

page line text 23 7

Corollary 0.2.20. If M = N_{1} + ... + N_{t} with each N_{i} Noetherian (resp. Artinian) then M is Noetherian (resp. Artinian).

79 4 Rather than define a self-dual category, we should say a class S of categories is self-dual if C^{op} ∈ S whenever C ∈ S.

99 -8 A_{1}(F) is not a PLID since the left ideal spanned by λ, μ is not principal.

198 14 Replace statement of lemma 2.5.50 by the following text:

If t(R_{0}) is quasi-invertible then |G|R_{0} is quasi-invertible. (Proof given in this paperback text.)

Some corrections for Remark 3.2.2. In (v), one should take subsets containing 0; (vii) is wrong and should be deleted. I thank Tazid Ali for pointing this out. (For example take R to be an infinite direct product of fields {F_{i}: i ∈ I}, and each A_{i} to be the i-th component, clearly an annihilator. But ∑ A_{i} is not an annihilator.)

397 -10 The uniform dimension is not additive (and indeed this was not claimed in 3.2.19).

415 -9 The proof of proposition 3.5.41 requires:

Remark 3.5.40′′. If M_{2} ≤ M_{1} ≤ M and N ≤ M then there is an exact sequence

0 → M_{1} ∩ N/ M_{2} ∩ N → M_{1}/M_{2} → (M_{1} + N)/(M_{2} + N) → 0

(Proof given in the Student Edition.)

VOLUME II. 18 -15 free abelian group

18 -11 in K_{0}(R), we see

65 Prop.5.3.15 central C-subalgebra C′

97 2,3 replace by the following text:

multilinear polynomial f is a weak identity of M_{n}(C) then T_{k}f is scalarvalued for all substitutions of matrices of trace 0.

126 10 replace (GK dim R_{1})(GK dim R_{2}) by GK dim(R_{1}) + GK dim(R_{2})

Typeset by AMS-TEX

126 -15 replace α_{1}α_{2} by α_{1} + α_{2}

126 -12 replace α_{i} by α_{1} (twice)

129 -7 prime affine

In the non-prime case, they show any affine algebra of GK dim 1 is PI, but need not be Noetherian. Still, by (Braun’s) theorem 6.3.39 below, the nilradical is nilpotent.

211 5 replace GK dim by GK tr deg

245,246 paragraphs misplaced - take lemma 7.3.6 through bottom of p. 246 and insert them between 7.3.3 and 7.3.4. This entails renumbering these three results and their references, which are used on pages 245 l. -3 and 247 lines 1,4,5,6,14,-12.

248 2 The argument relies on Corollary 7.2.11′ of the paperback text, which was not proved in the original volumes.

255 -11 τ does not send H_{q} to itself. However this can be remedied by using Lagrange resolvents, as noted by Tignol, cf. a forthcoming paper by Rowen and Tignol, which also includes a result for p = 2.

257 -9 There is a gap here, which could be filled for (odd) n = p^{4}.

353 -7 replace by the following text:

∑ r_{i}a^{i} of L∪T_{i} }. Note that the leading coefficient is not well- defined on T_{i}, but may vary according to the way that an element is written. However, it is well-defined modulo lead_{i}(0), and thus modulo lead_{i}(L) for any left ideal L.