Abstract algebra-> Let R be a ring and let M2(R) be the set of 2 x 2 matrices with

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cooljosh2k2
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Abstract algebra--> Let R be a ring and let M2(R) be the set of 2 x 2 matrices with

Homework Statement



Let R be a ring and let M2(R) be the set of 2 x 2 matrices with entries in R.
De fine a function f by:

f(r) = (r 0) <----matrix
........(0 r)

for any r ∈ R

(a) Show that f is a homomorphism.
(b) Find ker(f).

The Attempt at a Solution



I just want to know if this is right:

a) For any r, s in R,
f(r) + f(s) = (r 0)..+..(s 0)
.................(0 r)......(0 s) =

((r+s) 0)
(0 (r+s)) = f(r + s).

Therefore, f is a group homomophism.

b) ker f = {r ∈ R: f(r) = zero matrix}.

This is only possible if r=0.

Therefore, ker f = {0}
Hence, ker f = {0}
 

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