## New answers tagged counterexamples

2
votes

### Counterexamples in algebra?

A very basic one: Over the field of two elements, the symmetric matrix $\left(\begin{matrix}1&1\\1&1\end{matrix}\right)$
is nilpotent and thus not diagonalizable.

Community wiki

4
votes

### Counterexamples in algebra?

If $x$ and $y$ are elements of an associative ring such that $xy\ne1=yx$ then there is a mutually inverse pair of invertible matrices one of which is lower triangular but not upper triangular and the ...

Community wiki

1
vote

### Counterexamples in algebra?

OP: [...] counterexamples can illuminate a definition (e.g. a projective module that is not free), [...]
Indeed, let our ring $\ \mathcal R\ $ be the the ring of all continuous functions from the ...

Community wiki

1
vote

### Counterexamples in algebra?

You might find several answers in Harry Hutchins's book on Examples of Commutative Rings.

Community wiki

3
votes

### Counterexamples in algebra?

Matrices in $\text{Mat}_2(\mathbb{Z})$ not conjugate to their transpose by $\text{GL}_2(\mathbb{Z})$.
A matrix and its transpose are similar over any field (cf. here), thus a matrix $M\in\text{Mat}_2(\...

Community wiki

7
votes

Accepted

### Can a power series of several variables be discontinuous on a compact set if it converges in every point of this set?

The series $f(x,y)=y+xy+x^2y+x^3y+\dots$ converges to $0$ when $y=0$, and converges to $y/(1-x)$ when $|x|<1$. This function is not continuous at $(x,y)=(1,0)$.

Top 50 recent answers are included

#### Related Tags

counterexamples × 226gn.general-topology × 32

real-analysis × 24

nt.number-theory × 22

examples × 20

reference-request × 19

ct.category-theory × 16

prime-numbers × 15

fa.functional-analysis × 14

ag.algebraic-geometry × 13

graph-theory × 13

ac.commutative-algebra × 12

elementary-proofs × 12

gr.group-theory × 10

co.combinatorics × 9

pr.probability × 9

measure-theory × 8

ds.dynamical-systems × 8

at.algebraic-topology × 7

ca.classical-analysis-and-odes × 7

big-list × 7

ra.rings-and-algebras × 6

banach-spaces × 6

differential-equations × 5

inequalities × 5