Proof
We verify the three laws for I + J to be an ideal:
0 
I + J.
If u, u' I + J,
then u + u' I + J.
If r R
and
u I + J, then
ru I + J.
Clearly, 0 = 0 + 0 I + J.
I + J.
Suppose u, u' I + J.
Then there are
x, x' I and
y, y' J such that
u =
x + y and
u' = x' + y' J.
By commutativity of +, we find
u + u' = x + x' + y + y'
I + J.
I + J.
Then there are
x, x' I and
y, y' J such that
u =
x + y and
u' = x' + y' J.
By commutativity of +, we find
I + J.
Suppose r R
and
u = x + y I + J
with
x I and
y J.
Then
ru = rx + ry
I + J.
R
and
u = x + y I + J
with
x I and
y J.
Then
ru = rx + ry
I + J.