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This is certainly related to "What are your favorite instructional counterexamples?", but I thought I would ask a more focused question. We've all seen Counterexamples in Analysis and Counterexamples in Topology, so I think it's time for: Counterexamples in Algebra.

Now, Algebra is quite broad, and I'm new at this, so if I need to narrow this then I will- just let me know. At the moment I'm looking for counterexamples in all areas of algebra: finite groups, representation theory, homological algebra, Galois theory, Lie groups and Lie algebras, etc. This might be too much, so a moderator can change that.

These counterexamples can illuminate a definition (e.g. a projective module that is not free), illustrate the importance of a condition in a theorem (e.g. non-locally compact group that does not admit a Haar measure), or provide a useful counterexample for a variety of possible conjectures (I don't have an algebraic example, but something analogous to the Cantor set in analysis). I look forward to your responses!

You can also add your counter-examples to this nLab page: http://ncatlab.org/nlab/show/counterexamples+in+algebra

(the link to that page is currently "below the fold" in the comment list so I (Andrew Stacey) have added it to the main question)

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My feeling is that this question is far too broad. –  Andy Putman Jun 21 '10 at 23:11
I like that the question is general. I think if it's narrowed too much we won't get as many interesting responses. All of the big list type questions that have been successful have been fairly general, so I don't think it hurts as long as we aren't swarmed with questions like this. –  jeremy Jun 22 '10 at 0:23
Whilst I like lists of counterexamples, I don't think that MO is an appropriate place for one. I've explained why in the meta discussion (NB: please vote for the comment linking to the meta discussion so that it appears "above the fold"). I think that this would work so much better as a wiki page. So I've started one on the nLab: ncatlab.org/nlab/show/counterexamples+in+algebra Obviously, as I'm not an algebraist I didn't understand everything and have probably left out a lot of information in copying it over. I recommend closing this question and redirecting to that nLab page. –  Loop Space Jun 22 '10 at 8:33
Andrew, why not keep the question open, in order to generate the examples here that can then be more sensibly organized on your page? It seems likely to me that you will get a lot of good examples with this question that might otherwise be missed. –  Joel David Hamkins Jun 22 '10 at 13:16

45 Answers 45

An infinitely-generated Noetherian ring: $\mathbb{Q},$ the field of rational numbers.

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A finitely generated module with a non-finitely generated submodule: Consider the polynomial ring $k[x_1, x_2, ...]$ as a module over itself. The submodule generated by $\{ x_1, x_2, ...\}$ is not finitely generated.

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While a finite abelian group is determined by its character table, this is not true for (finite) nonabelian groups. E.g., the dihedral and quaternion groups of order 8 (or more generally two nonabelian groups of order p3 for a prime p) are nonisomorphic but have the same character table.

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Regarding Schur's lemma:

For a finite group $G$ and $V$ a finite-dimensional irreducible representation of $G$ over a field $K$, there exist endomorphisms of this representation that are not scalar multiples of the identity. For example, take $G=\mathbb{Z}_4$, $K=\mathbb{R}$, and $\rho:\mathbb{Z_4}\rightarrow GL(\mathbb{R}^2)$ given by

$$\rho(1)=\left(\begin{align} 0 & -1 \\\ 1 & \ \ 0 \end{align}\right)$$

Then since $\rho(1)$ has no real eigenvalues the representation is irreducible. But on the other hand, $\mathbb{Z}_4$ is abelian and $\rho(1): \mathbb{R}^2\rightarrow\mathbb{R}^2$ is an endomorphism of this representation.

This is why it is important $K$ be algebraically closed.

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If $V$ is any representation of anything, and if $M$ is any matrix without eigenvalues in $K$ that commutes with everything, then it is a counterexample to Schur's lemma without algebraic closure (the proof of the lemma tells you this). In particular, if the group is abelian these are easy to find. What if the group is not abelian? –  Ryan Reich Jun 20 '11 at 15:42

The quaternion group of order $8$ has a real irreducible character of degree $2,$ but the associated representation can not be realized over the real field.

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A subring of a UFD need not be a UFD.

An example by M. Zafrullah: Let R be the set of real numbers and Q be the set of rational numbers. Then the polynomial ring R[X] is a UFD (since it is a PID), but its subring Q + XR[X] is not a UFD.

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@Pete: Your example is simple in a writing-down-sense. $\mathbb{Z}[\alpha]$ may well be the shortest expression for such a ring. If one is intressted in proof as well, then your example is much more complicated than $\mathbb{Q}+X\mathbb{R}[X]$ because one has to define and prove several things about algebraic integers first. –  Johannes Hahn Aug 11 '10 at 9:47
Every integral domain is a subring of a UFD (for example, its field of fractions). So all you need here is an integral domain which is not a UFD. –  Chris Eagle Jun 18 '11 at 8:50

Desmond MacHale wrote an article, "Minimal Counterexamples in Group Theory", Mathematics Magazine, 54 (1981), no. 1, 23–28. I've found this paper useful in an introductory algebra class and I like the philosophy of the paper, "Is X true? No, probably not. So what is a smallest counterexample?" A variation on the group theory (and Irish!) tune of MacHales appears here. A followup article is "Constructing a minimal counterexample in group theory" by Arnold Feldman, also in Mathematics Magazine (1985).

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Radical of a primary ideal is prime but not every ideal whose radical is prime is primary. Here is a cute counterexample: Let $I=(x^2,xy)\in F[x,y]$ where $F$ is a field. The radical $\sqrt{I}$ of $I$ is $(x)$ which is prime but $I$ is not primary; $xy\in I$, $x\not\in I$ but no power of $y$ belongs to $I$.

This is from page 154 of Commutative Algebra Vol. 1 by Zariski and Samuel. Now that I check, this is the 1975 printing which I bought on 1979. How time flies when you are having fun! :-)

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My favorite counter example in Galois theory is the field extension $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$.

Here are some cases to which it provides a counter example:

  1. It is a non-Galois extension (so providing counter example to "every extension is Galois").

  2. The Galois group of its Galois closure is non-abelian.

  3. Although the intersection of it with $\mathbb{Q}(\zeta_3)$ is $\mathbb{Q}$, these fields are not linearly disjoint.

[Here $\zeta_3=e^{\frac{2\pi i}{3}}$.]

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An example showing that, in the standard definition of a ring (without $1$), it won't do to replace the left and right distributive laws with the single law$$(u+v)\cdot(x+y)=u\cdot x+u\cdot y+v\cdot x+v\cdot y$$as was done on p. 18 of my old copy (July 1957 printing) of Kelley's General Topology.

(Spoiler alert: if you hover over the gray box, an example of what can go wrong is revealed.)

Take an additive group of order $3$, choose a nonzero element $c$, and define $x\cdot y=c$.

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Assertion: If $S$ is an associative ring with identity and $M, N$ are unital left $S$-modules, then $\hom_S(M,N)$ is a unital left $S$-module.

Th is is false in general. Consider the matrix ring $S:=M_2({\mathbb Z})$, and let $M={\mathbb Z}^2=N$ be equipped with natural $S$-module structure. We have $\hom_S(M,N)\hookrightarrow S$, as additive groups; moreover, if $\phi\in \hom_S(M,N)$, then $\phi(A{\bf v})=A(\phi({\bf v}))$ for every $A\in M_2({\mathbb Z})$ and ${\bf v}\in M$. Thus, the scalar matrices, and only those, are the elements in $\hom_S(M,N)$, since the only matrices that commute with all four of the elementary matrices are precisely the scalar matrices. By the isomorphism ${\mathbb Z}\cong {\mathbb Z} I_2$, we have $X:=\hom_S(M,N)\cong {\mathbb Z}$, and this makes $\hom_{\mathbb Z}(X,X)\cong {\mathbb Z}$.

Now, to make $X=\hom_S(M,N)$ into a unital left $S$-module, we need to give a unital ring homomorphism $S\rightarrow \hom_{\mathbb Z}(X,X)\cong {\mathbb Z}$. But there is no such. To see this, we notice that the elementary matrices in $S$ are nilpotent elements, and therefore, must be mapped to the unique nilpotent element $0\in {\mathbb Z}$. Since a ring homomorphism is additive, the only homomorphism $S\rightarrow {\mathbb Z}$ must be trivial.

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If $f$ and $g$ are relatively prime in ${\mathbf Q}[X]$ then the mapping ${\mathbf Q}[X]/(fg) \rightarrow {\mathbf Q}[X]/(f) \times {\mathbf Q}[X]/(g)$ given by $h \bmod fg \mapsto (h \bmod f, h \bmod g)$ is a ring isomorphism. This is a special case of the Chinese remainder theorem.

If we replace ${\mathbf Q}[X]$ with its subring ${\mathbf Z}[X]$, which is a UFD, then for relatively prime $f$ and $g$ in ${\mathbf Z}[X]$ the mapping ${\mathbf Z}[X]/(fg) \rightarrow {\mathbf Z}[X]/(f) \times {\mathbf Z}[X]/(g)$ given by $h \bmod fg \mapsto (h \bmod f, h \bmod g)$ is a ring homomorphism, but it is not necessarily an isomorphism since it need not be surjective (though it is injective). For example, if $f = X-1$ and $g = 1+X+\cdots + X^{n-1}$ where $n > 1$ then the natural mapping $${\mathbf Z}[X](X^n-1) \rightarrow {\mathbf Z}[X]/(X-1) \times {\mathbf Z}[X]/(1+X+\cdots + X^{n-1})$$ does not have $(0,1)$ in its image.

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If $K/\Bbb Q$ is a number field and $\mathcal{O}_K$ its ring of integers, $\mathcal{O}_K$ need not have a power basis as a $\Bbb Z$-module; i.e., $\mathcal{O}_K\neq\Bbb Z[\alpha]$ for any $\alpha\in\mathcal{O}_K$! An example is given by $K = \Bbb Q(\alpha)$, where $\alpha$ is a root of $T^3 - T^2 - 2T - 8$ (a $\Bbb Z$-basis is instead given by $\{1,\alpha,(\alpha^2 + \alpha)/2\}$).

See Keith Conrad's notes for more detail on this example.

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Here's one from universal algebra: the class of mono-unary algebras $\mathfrak A=(A,f)$ such that $f$ is a permutation with a unique fixed point does not have the unique factorization property for direct decomposition.

(Spoiler alert: if you hover over the gray box, an example of what can go wrong is revealed.)

For $n\equiv1\pmod3$ let $\mathfrak A_n=(A,f)$ where $|A|=n$ and $f$ is a permutation of order $3$ with a unique fixed point; then $\mathfrak A_{100}\cong\mathfrak A_{10}\times\mathfrak A_{10}\cong\mathfrak A_4\times\mathfrak A_{25}$, and $\mathfrak A_4,\mathfrak A_{10},\mathfrak A_{25}$ are indecomposable.

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