Proof that there is no ring homomorphism between M2(R) [2x2 matrices with real elements] and R2 (normal 2-dimensional real plane).
The Attempt at a Solution
I have tried to proof this problem with properties of ring homomorphism (or finding such a property of ring homomorphism that doesn't fill in this situation. I have tried nilpotents, inverse elements, etc.)
This would be easy one if those two rings were integral domains. In general speaking, how to proof that there does not exist any ring homomorphism between two rings when the rings aren't integral domains?