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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.
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 ...
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.
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(\...
7 votes
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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 $(x,y)=(1,0)$.
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