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

1. Feb 4, 2010

### gotmilk04

1. The problem statement, all variables and given/known data
Give an example of a ring monomorphism f:M(2;R)-M(3;R)

2. Relevant equations

3. The attempt at a solution
I can't think of anything that would be a monomorphism.

Last edited: Feb 4, 2010
2. Feb 5, 2010

### 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?