Is there a good errata for AtiyahMacdonald available? A cursory Google search reveals a laughably short list here, with just a few typos. Is there any source available online which lists inaccuracies and gaps?

Dear Tim, on page 31 they consider a ring $A$ and two $A$ algebras defined by their structural ring morphisms $f:A\to B$ and $g:A\to C$. They then define the tensor product as a ring $D=B\otimes _A C$ and want to make it an $A$ algebra. For that they must define the structural morphism $A\to D$ and they claim that it is given by the formula $a \to f(a)\otimes g(a)$.This is false since that map is not a ring morphism. The correct structural map $A\to D$ is actually $a\mapsto 1_B\otimes g(a) =f(a)\otimes 1_C$. PS: To prevent misunderstandings, let me add that AtiyahMacDonald is, to my taste, the best mathematics book I have ever seen, all subjects considered. 


Nearly all the mistakes pointed out so far were fixed in the Russian translation, which was done by Manin. But not all. I'll list in parentheses the page numbers of the translation where the original error still occurs for the 5 people who might care. (The translation is usually 11 page numbers ahead of the original.) Scan the answers posted before this one to determine which mistakes I am referring to. p. 29 (> p. 41): on line 8, change (2.14) to (2.13) p. 55 (> p. 66): exercise 2 p. 71 (> p. 82): exercise 23 p. 88 (> p. 99): exercise 27(v) There were also completely original mistakes added especially for the translation! On page 30 line 7 and page 31 lines 10 and 14 of the translation, the tensor product signs should be direct sum signs. On page 32 in the statement of Nakayama's Lemma, the ideal a should be in fraktur font. 


EDIT OF JUNE 9, 2011 Page 102, penultimate paragraph: "... $f$ induces a homomorphism $\widehat{f}:\widehat{G}\to\widehat{H}$, which is continuous." No topology has been defined on $\widehat{G}$ and $\widehat{H}$. [July 7, 2011, GMT. The topology on $\widehat{G}$ can be described as follows. For any subset $S$ of $G$, let $\widehat{S}\subset\widehat{G}$ be the set of equivalence classes of Cauchy sequences in $S$, and say that a subset $V$ of $\widehat{G}$ is a neighborhood of $0$ if there is a neighborhood $W$ of $0$ in $G$ such that $\widehat{W}\subset V$.] By the way, there is (I think) a somewhat similar "mistake" in the article Atiyah wrote with Wall in "Algebraic Number Theory" Ed. Cassels and Froehlich (see Erratum for CasselsFroehlich). Atiyah and Wall forgot to mention the crucial compatibility between change of groups and connecting morphisms. (See p. 99.) END OF EDIT OF JUNE 9, 2011 Page 25, first line of the proof of (2.13): change (2.11) to (2.12). Page 29, about two third of the page: change (2.14) to (2.13). EDIT. Page 39, last line: change $m$ to $m_i$ (three times). EDIT OF NOV. 22, 2010. Page 63, proof of Lemma 5.14. The current text reads Conversely, if $x\in r(\mathfrak a^e)$ then $x^n=\sum a_i\,x_i$ for some $n>0$, where the $a_i$ are elements of $\mathfrak a$ and the $x_i$ are elements of $C$. Since each $x_i$ is integral over $A$ it follows from (5.2) that $M=A[x_1,\dots,x_n]\ \dots$ It would be better (I think) to write something like Conversely, if $x\in r(\mathfrak a^e)$ then $x^n=a_1\,x_1+\cdots+a_m\,x_m$ for some $m,n>0$, where the $a_i$ are elements of $\mathfrak a$ and the $x_i$ are elements of $C$. Since each $x_i$ is integral over $A$ it follows from (5.2) that $M=A[x_1,\dots,x_m]\ \dots$ [July 8, 2011, GMT. Page 90. It seems to me that the second part of the proof of Theorem 8.7 can be simplified. We must check the uniqueness of the decomposition of an Artin ring $A$ as a finite product of Artin local rings $A_i$. To do this it suffices to observe that, for each minimal primary ideal $\mathfrak q$ of $A$, there is a unique $i$ such that $\mathfrak q$ is the kernel of the canonical projection onto $A_i$.] [July 7, 2011, GMT. Page 107, lines 45. Instead of $A^*=A[x_1,\dots,x_r]$ read $A^*=A[y_1,\dots,y_r]$ where $y_i=(0,x_i,0,\dots)$.] [July 7, 2011, GMT. Page 112, proof of Proposition (10.24). Instead of $\mathfrak{a}^{k+n(i)}$ read $\mathfrak{a}^{\max(0,kn(i))}$.] [July 9, 2015. The integer $d(M)$ (and in particular $d(A)$) is defined on p. 117 after the proof of Theorem 11.1. Another definition of $d(A)$ is given on p. 119 after the proof of Proposition 11.6 via the equality $d(A)=d(G_{\mathfrak m}(A))$. But the old meaning of $d(?)$ is used again in the proof of Proposition 11.20 p. 122, where the expression $d(G_{\mathfrak q}(A))$ occurs at the beginning of the last display. To avoid any confusion, let me denote by $D(M)$ the integer given by the first definition, and set $d(A):=D(G_{\mathfrak m}(A))$. It seems to me the proof of Proposition 11.3 p. 118 is not entirely correct. I suggest to keep the proof, but to weaken slightly the statement, the new statement being: If $P(M/xM,t)\neq0$ and $D(M/xM)\ge1$, then $P(M,t)\neq0$ and $D(M/xM)=D(M)1$. This new statement applies to the first equality in the last display in the proof of Proposition 11.20 p. 122 if $d:=\dim A\ge1$ (the case $d=0$ being trivial).  On the third line of the proof $\mathfrak q$ should be $\mathfrak q^2$.] 


On page 8, the proof of part ii of Proposition 1.11 begins "Suppose $\mathfrak{p}\not\subseteq\mathfrak{a}_i$ for all $i$." It should be $\not\supseteq$. 


On page 29, the example at the top has two typos: it says "$(x)=2x$", when it should be "$f(x)=2x$", and the exact sequence at the end of that same line says "$0\rightarrow\mathbb{Z}\otimes \stackrel{f\otimes 1}{\longrightarrow} \mathbb{Z}\otimes N$", when it should be "$0\rightarrow\mathbb{Z}\otimes N\stackrel{f\otimes 1}{\longrightarrow} \mathbb{Z}\otimes N$". 


On p.55, exercise 4.2 reads "If $\mathfrak a = r(\mathfrak a)$, then $\mathfrak a$ has no embedded prime ideals". I believe it should include the assumption that $\mathfrak a$ is decomposable. AM defines embedded primes for decomposable ideals only. And it doesn't seem that a radical ideal should automatically be decomposable. If you take something like a reduced (nonnoetherian) ring with infinitely many minimal prime ideals, I expect the zero ideal will be radical but not decomposable... 


Minor typos: p.34, exercise 2.23: Second sentence should start "For each finite subject $J$ of $\Lambda$". p.48, exercise 3.27(i): The bracketed text should read "Use Exercises 25 and 26". p.71, exercise 5.23: The hint should start "The only hard part is (iii) => (i). Suppose (i) is false". p.88, exercise 7.27(v): The last clause should read "the homomorphism $f_{!}$ is a $K_1(A)$module homomorphism". p.127, index entry for "flat, faithfully": Should cite p. 46, not p. 29. 


page 81, line 5: change $f_i \in A[x]$ to $f_i \in \mathfrak{a}$ 


Also minor: On p. 91, the $a$'s and $\mathfrak a$'s in the proof of Prop 8.8 seems to be a little jumbled. I guess you want something like "Let $\mathfrak a$ be an ideal of $A$, other than $(0)$ or $(1)$. We have $\mathfrak m = \mathfrak N$, hence $\mathfrak m$ is nilpotent by (8.4) and therefore there exists a positive integer $r$ such that $\mathfrak a \subseteq \mathfrak m^r$ and $\mathfrak a \not\subseteq \mathfrak m^{r + 1}$; hence there exists $y \in \mathfrak a$ and $a \in A$ such that $y = ax^r$ but $y \not\in (x^{r + 1})$," etc. 


Page 114, Exercise 5, the short exact sequence is missing the middle term. 


On page 41 in the proof of proposition 3.10., change "i) $\implies$ ii) by (3.5) and (2.20)" to "i) $\implies$ ii) by (3.7) and (2.19)" On page 52 in remark 1) at the bottom of the page, change "(see Chapter 1, Exercise 25)" to "(see Chapter 1, Exercise 27)" On page 65 at the end of the proof of proposition 5.18. the black square to denote end of proof is missing. On page 66 we need to correct the proof of corollary 5.22., one correct version is the following: We start with the quotient map $\pi: A[x^{1}] \to A[x^{1}] /m$ where $m$ is a maximal ideal containing $x^{1}$. We take an algebraic closure $\Omega$ of the field $A[x^{1}] /m$ and consider the map $i \circ \pi: A[x^{1}] \to \Omega$. Then by the previous theorem, (5.21), we can extend $i \circ \pi$ to some valuation ring $B$ of $K$ containing $A[x^{1}]$: $g: B \to \Omega$ such that $g_{A[x^{1}]} = i \circ \pi$. Then $g(x^{1}) = 0$. Hence $x^{1} \in ker(g)$ and since the kernel is a proper ideal of $B$, $x^{1}$ is not a unit in $B$ and hence $x$ is not in $B$. (also see math.SE) On page 77 in the proof of proposition 6.7., change "...a composition series, by ii);..." to "...a composition series, by i);..." 


On page 31, the first line refers to Proposition 2.11, when it should be 2.12. 


On page 91, the second line in the second Example should refer to Proposition 8.8, not Theorem 8.7. 


On page 23, in the third line of the sketch for Proposition 2.9, change "$v \circ u \circ f = 0$ " to "$f \circ v \circ u = 0$". 


In p.45, Ex.3.12.iv, one can avoid the tedious argument provided in the hint by noting that $K\otimes_A M\cong(A\{0\})^{1}M$. (I originally did it as hinted ...) In p.68, Ex.5.10.ii, (b') is actually equivalent to a weaker (c') that asserts only that $f^*:\mathrm{Spec}(B_\mathfrak{q})\to\mathrm{Spec}(A_\mathfrak{p})\cap V(\ker f)$ is surjective. However, (a') does imply the original (c'). 


Here are a few more small miscellaneous mistakes and typos: page 18, line 6: $M''$ not defined (it is $M/M'$) page 20, line 12: in the expression for $A$ as a direct product, it should be $i=1$ not $i=I$. page 28, line 5: $=$ should read $\cong$ page 29, line 13 (first line of proof that iv) $\Rightarrow$ iii)): $x_{i}$ should read $x_{i}'$ page 91, in the last example, it is not true that ${\mathfrak m}^{2}=0$. It is a nonzero principal ideal. But the following statement is still true, $\dim \left( {\mathfrak m}/{\mathfrak m}^{2} \right)=2$. It is generated by $x^2$ and $x^3$. page 102, Lemma 10.1(iv) $H=0$ should read $H=\{ 0 \}$ 


On p.89, the second to last line of the proof of Proposition 8.4 should say $\mathfrak{N}^k \subseteq \mathfrak{ N}$ instead of $\mathfrak{N}^k \supseteq \mathfrak{N}$. 


Page 69, Ex5.17: this is not the weak form, and the result is rather trivial. 


Page 118, line 16, Example: Poincaré series should be $P(A,t)=(1t)^{s}l(A_0)$. 


Here are a few more minor errors: p. 42, proof of Prop. 3.11(iv), last line: change "by i)" to "by ii)". p. 72, Exercise 29 (clarification): by "a local ring of $A$" they mean "a localization of $A$ at some prime ideal". (See this question.) p. 72, Exercise 31: for condition (2), change "for all $x,y \in K^*$" to "for all $x,y \in K^*$ such that $x+y \neq 0$". (Similarly on p. 94, unless one uses the convention $v(0)=+\infty$.) p. 74, Example 2: change "if $a \neq 0$" to "if $a \neq 0$ and $a \neq \pm 1$". p. 75, proof of Prop. 6.2: change "hence $M_n=M$" to "hence $M_n=N$". p. 82, definition of irreducible ideal: it is understood (by the first sentence of the section) that an irreducible ideal $\mathfrak{a}$ is also required to be proper (otherwise Lemma 7.12 would be false, for example). Also, there should perhaps be an entry "Ideal, irreducible, 82" in the index on p. 127. p. 94, paragraph before Prop. 9.2: change "$v \colon K^* \to Z$" to "$v \colon K^* \to \mathbf{Z}$". p. 111, definition of $G(A)$ in the middle of the page: change $a^n$ to $\mathfrak{a}^n$. And finally some really important corrections... ;) p. 30, middle of the page: missing comma in "finite set of elements $x_1,\ldots x_n$". p. 61, proof of Prop. 5.6(ii): missing space before the brackets in "$x/s \in S^{1}B(x \in B,\, s \in S)$". p. 83, page header: change "NEOTHERIAN" to "NOETHERIAN". 


Here is what I found (in LaTeX unlike in the original); only two of the errors mentioned below (both indicated) have been reported in previous answers here. http://web.archive.org/web/20090822170221/http://math.mit.edu/~ssam/soln/commalg.errata p.56, line 24 (Exercise 13(iv)): change second ${\mathfrak p}^{(n)}$ to ${\mathfrak p}^n$ p.76, line 11: change "Exercise" to "Example" p.76, line 14: change "Exercise" to "Example" p.91, line 11 (line 2 of second example): change (8.7) to (8.8) p.97, line 1: change (9.7) to (9.6) p.104, line 4: change $\left\{ G'_n \cap G_n \right\}$ to $\left\{ G' \cap G_n \right\}$ p.114, line 12: change $0 \rightarrow {\mathfrak b}^{m} M \rightarrow M/{\mathfrak b}^{m} M \rightarrow 0$ to $0 \rightarrow {\mathfrak b}^{m} M \rightarrow M \rightarrow M/{\mathfrak b}^{m} M \rightarrow 0$ 


Proof of Theorem 11.22, page 123, direction $iii) \Rightarrow i)$: The map should be $\alpha : k[t_1,\dots,t_d] \rightarrow G_{\mathfrak{m}}(A)$. 


p. 107, lines 45: change "$A^* = A[x_1,\dots,x_r]$" to "$A^*$ is a quotient of $A[x_1,\dots,x_r]$". 


Maybe in page 22, in the proof of corollary 2.7, = should be ≅? 

