Prove or disprove that this is a ring homomorphism.

Consider the mapping from M2(Z) into Z given by the matrix
[a b]
[c d] --> a.

Prove or disprove that this is a ring homomorphism.

Thanks for your help

This is not a ring homomorphism, because it violates ring multiplication.

Example:
[2 1] [3 1]
[1 2],[1 3], map to 2, 3 (respectively) with product 6, but

[2 1][3 1]....[7 5]
[1 2][1 3] =.[5 7], which maps to 7.

I hope this helps!