# Tag Info

### 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.

### 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 ...
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 ...
1 vote

### Counterexamples in algebra?

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