Ring monomorphism from M(2;R)-M(3;R)

[gotmilk04]

Homework Statement

Give an example of a ring monomorphism f:M(2;R)-M(3;R)

Homework Equations

The Attempt at a Solution

I can't think of anything that would be a monomorphism.

 

[ystael]

I assume you mean the rings $$M_2(\mathbb{R})$$ and $$M_3(\mathbb{R})$$ of respectively $$2 \times 2$$ and $$3 \times 3$$ matrices with real entries.

Think of a monomorphism as an "embedding", i.e., how can you embed the $$2\times 2$$ matrices in the $$3\times 3$$ ones without changing their structure?

One way to attack this is to use geometry. A $$2\times 2$$ matrix $$A$$ is a linear transformation of $$\mathbb{R}^2$$. How can you extend $$A$$ to be a linear transformation of $$\mathbb{R}^3$$, in such a way that you don't "interfere with" the action of $$A$$ on $$\mathbb{R}^2$$ in any way?