- #1

cooljosh2k2

- 69

- 0

**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.

Define 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}