This is probably simple. Is there a finite noncommutative ring $R$ with identity in which all of its ideals are twosided !?

$\begingroup$ For a vector space $V$ of dimension $\geq 2$ over a finite field $F$, have you considered the exterior algebra on $V$? $\endgroup$– Christopher DrupieskiCommented Jan 31, 2013 at 21:01

$\begingroup$ Sorry, this is not the same as mathoverflow.net/questions/117561/apropertyoffiniterings but closely related. (I deleted a comment saying it is a duplicate.) $\endgroup$– Benjamin SteinbergCommented Jan 31, 2013 at 22:31
2 Answers
A "natural" example is given by the group ring $\mathbb{F}_2[Q]$ of the Quaternion group of order 8.
For, we have to show that each left ideal is also a right ideal, and conversely, each right ideal is also a left ideal. The first half (i.e. left is right) is shown in this paper.
Let $i:\mathbb{F}_2[Q] \to \mathbb{F}_2[Q],\;g \mapsto g^{1}$ be the antipode. It's a general fact that for a left (right) ideal $I$, $i(I)$ is a right (left) ideal.
Now suppose $I$ is a right ideal. Hence $i(I)$ is a left ideal and by the above, it's also a right ideal. Consequently, $I=i(i(I))$ is a left ideal and we are done.
Added: The comment asks for a modular representation of $Q$. Using GAP I found that $\mathbb{F}_2[Q]$ can be embedded into the matrix ring $M_4(\mathbb{F}_2)$. Write $Q=\langle x,y\mid x^4=y^4=1, yxy^{1}=x^{1}\rangle$. Then a faithful representation $Q\hookrightarrow GL(4,2)$ is given by $$x \mapsto \begin{pmatrix}1 & 0 & 1 & 0 \newline 0 & 1 & 0 & 0 \newline 0 & 0 & 1 & 1 \newline 0 & 0 & 0 & 1 \end{pmatrix}\qquad y \mapsto \begin{pmatrix}1 & 1 & 1 & 1 \newline 0 & 1 & 0 & 1 \newline 0 & 0 & 1 & 0 \newline 0 & 0 & 0 & 1 \end{pmatrix}$$ Since the Sylow 2subgroup of $GL(3,2)$ is the Dihedral group $D_8$, four is the smallest degree of a faithfull representation of $Q$ over $\mathbb{F}_2$.

$\begingroup$ @Ralh: Thanks Ralph. Can you represent $\mathbb{F}_2[Q]$ as (perhaps subrings or some products) of matrix rings ? $\endgroup$– user30338Commented Feb 1, 2013 at 5:18

$\begingroup$ @Dimros: every finite dimensional algebra can be represented as a subalgebra of a matrix algebra! $\endgroup$ Commented Feb 1, 2013 at 18:58

$\begingroup$ @Mariano: Yes, but usually one is interested in a faithful representation of preferably low degree. For example, in the case of the Quaternion group, the $4\times 4$matrices given above are considerably easier to work with than, say, the $8\times8$matrices arising from the regular representation. $\endgroup$– RalphCommented Feb 1, 2013 at 19:59
For a finite field $k$ with nontrivial automorphism $\sigma$, take the skew polynomial ring $k[X, \sigma]$ (reminder: these are polynomials with coefficients on the left $\sum a_i X^i$ with relation $Xa = \sigma(a)X$) and set $R := k[X, \sigma] / < X^n >$ (i.e. divide out the (twosided) ideal generated by $X^n$).
The only left as well as right ideals in this ring are the twosided ones, namely, the ones generated by one of the $X^k$ for $0 \le k \le n$. The key fact to see this is that the units of $R$ are precisely the polynomials $\sum a_i \bar X^i$ with $a_0 \neq 0$ (to see this, use e.g. a geometric series argument). In case $n=2$, it is easily written down explicitly, $R$ can be seen as the set of $a + bX$ with $a, b \in k$ and multiplication given by
$(a+bX)(c+dX) = ac + (b\sigma(c) + ad) X$;
for $a \neq 0$, we have $(a + bX)^{1} = a^{1}  a^{1} \sigma(a^{1}) b X$.
@ Dimros, concerning your questions to Ralph: A matrix ring $M_n(k)$ over a finite (division) field $k$ (and a fortiori, any product of these) will never meet your conditions, as for $n =1$ it is commutative and for $n >1$ it has left ideals which are not twosided. So by ArtinWedderburn, any example will have nonzero radical and its semisimple quotient will be a product of fields. On the other hand, every ring embeds into any matrix ring over itself, so every example is a subring of matrix rings. Maybe you want some special kind of subring.