Help with abstract algebra please, 10 points right away

Let L be the ring of all matrices in M_2(R) of the form of this 2x2 matrix:
(a 0)
(b c)

Show that the function f : L → R x R given by f (matrix shown above) = (a,c) is a surjective homomorphism, but not an isomorphism.

Prove 3 points:
1) f is a homomorphism
2) f is surjective (i.e. onto)
3) f is not injective (i.e. not one-to-one)

Homomorphism:
f[(a 0)]+ f[(a' 0)] =
[(b c)]....[(b' c')]

(a,c) + (a',c') =
(a + a',c + c') =

f[(a+a' 0....)]
[(b+b' c+c')]

as desired.

Surjective: for any (a,c)
(a,c)

f[(a 0)]
[(0 c)]

not injective:
f[(1 0)]
[(0 1)] =

f[(1 0)]
[(1 1)]

QED