Part 1
Suppose A and B are two subrings such that
A+B is a subring if B⊆A
∴∃a∈A such that a∈/B
∃b∈B such that b∈A
A+B satisfies the axioms
(i)A+B=∅
0R∈A and 0R∈B
⟹A+B is not empty
(ii)
a−b∈A+B
If a−b∈A then a−(a−b)∈A⟹b∈A
(iii)
∀a,b∈A+B,a.b∈A+B
Part 2
A is a subring of R ⟹A=∅
If a,b∈A then a−b∈A+B
If r∈R and a∈A+B then ar,ra∈A+B
⟹A is an ideal of A+B
Part 3
(i)
A,B=∅
⟹AnB=∅
(ii)a,b∈AnB⟹a−b∈AnB
(iii)r∈R,b∈AnB⟹br,rb∈AnB
Therefore AnB is an ideal of B
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