# Finding an isomorphic subring of matrices

I'm struggling a fair amount with this exercise:

Find a subring of $$M(2,\mathbb{Q})$$ which is isomorphic to a) $$\mathbb{Q}$$ x $$\mathbb{Q}$$ b) $$\mathbb{Q}$$ c) $$\mathbb{Q}[x]$$/$$x^2$$

Now I know a subring must be a subgroup, must contain the elements $$0,1$$ and must be closed under multiplication. As we are looking for isomorphisms then they must also be ring homomorphisms and must be bijective.

I tried to come up with a random subring and attempt to prove it is an isomorphism. E.g $$\left\{\left.\begin{matrix} a & b \\ c & 0 \end{matrix}\right|a,b,c \in \mathbb{Q}\right\}$$ is a subgroup of $$M(2,\mathbb{Q})$$ and then try and show it's an isomorphism. However I'm stuck here as I don't actually know how to go about finding these specific isomorphisms and I completely lose what I'm doing. Any help would be great.

• "got very confused with the form of answer the question needs" - what exactly is your confusion? Terminology, notation, etc? – lisyarus Feb 26 '19 at 12:54
• Welcome to MSE. Please show your attempts and efforts to solve this question. – Alan Muniz Feb 26 '19 at 12:56
• – lhf Feb 27 '19 at 13:53
• Why do you ask the same question again and again? – Dietrich Burde Feb 27 '19 at 15:17

Yes, we're talking about a ring homomorphism.

If we're looking for something isomorphic to $$\mathbb{Q}$$, note that we can get the integers by adding up copies of $$1$$. Then adding $$n$$ copies of $$\frac1n$$ gets us $$1$$; we're going to need all the scalar multiples of whatever we're mapping to $$1$$.

So, the subring we want to map to $$\mathbb{Q}$$ will consist of the scalar multiples of some $$A$$, and that $$A$$ will be mapped to $$1$$. Now, it's time to look at multiplication. What can we say about multiplication by $$A$$?

• ah okay that makes a lot more sense thank you, would it be very similar then when checking if its isomorphic to another ring, like $\mathbb{Q}$ x $\mathbb{Q}$ – L G Feb 26 '19 at 13:01
• Similar, yes. Any ring homomorphism will also be a $\mathbb{Q}$-linear map. – jmerry Feb 26 '19 at 13:09

Hints: Start with part b), and take the scalar multiples of the identity matrix.
For a), consider the diagonal matrices.
Finally for c), map $$1$$ to the identity matrix and map $$x$$ to a nontrivial matrix $$M$$ that satisfies $$M^2=0$$.

• Thanks a lot for the hints, but if i wasn't given those, how would one normally come about thinking that up? – L G Feb 27 '19 at 13:49

Hint: Write the three rings in the form $$\{ a + b u : a,b \in \mathbb Q \}$$ with $$u$$ satisfying a quadratic equation, $$u^2=cu+d$$. Then identity the ring with a matrix subring using the matrix representation: $$a+bu \leftrightarrow \pmatrix{ a & bd \\ b & a+bc}$$ This representation comes from the map $$z \mapsto (a+bu)z$$ in the basis $$1,u$$.

• – lhf Feb 27 '19 at 0:27
• Thank you very much, I will try it – user1230 Feb 27 '19 at 22:56