Intersection of Subrings

Theorem: The intersection of two subrings is a subring.

Let {S_1} and {S_2} be two subrings of ring R.
Since 0 \in {S_1} and 0 \in {S_2} at least 0 \in {S_1} \cap {S_2}. Therefore {S_1} \cap {S_2} is non-empty.
Let a,b \in {S_1} \cap {S_2}, then
a  \in {S_1} \cap {S_2} \Rightarrow a \in {S_1} and a \in {S_2}
b  \in {S_1} \cap {S_2} \Rightarrow b \in {S_1} and b \in {S_2}
But {S_1} and {S_2} are subrings of R, therefore
a,b  \in {S_1} \Rightarrow a - b \in {S_1} and ab \in {S_1}
a,b  \in {S_2} \Rightarrow a - b \in {S_2} and ab \in {S_2}
Consequently, a,b \in  {S_1} \cap {S_2} \Rightarrow a - b \in {S_1} \cap {S_2} and ab \in {S_1} \cap {S_2}.
Hence, {S_1} \cap  {S_2} is a subring of R.