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?