Let R be a ring

Suppose that I_a is an ideal of R, 1=< a=< n.

Show that: I_1 + I_2+....+ I_n = {i_1 + i_2+...+ i_n: a in I, 1=< a=< n} is an ideal of R.

This is my attempt

Since 1 belongs to R then 1 is also belongs to I_a implies I_a is non empty set

For any i_1, i_2 in I_a, then i_1 + i_2 also belongs to I_a

So, for 1=< a=< n, i_1 +.....+ i_n belongs to I_a

Hence, (I_1 + I_2+...+ I_n) is an ideal of R

Is that correct?

Thank you