Ideal Generated by two elements (Notation question)

Solution 1:

Assuming you mean $R$ to be commutative or ideal to be two-sided: Yes, it is: by the definition, $(r,s)=\{rx+sy|x,y\in R\}$.
On the other hand, $(r)+(s)=\{ax+by|a\in(r),b\in(s),x,y\in R\}$, which is the same since $a\in(r)$ implies that $a=r\alpha$.

Solution 2:

By definition, $(r,s)$ is the smallest ideal containing both $r$ and $s$.

By definition, $(r)+(s)$ is the sum of the ideals generated by $r$ and $s$.

That these two are equal is actually an easy theorem.

More generally, if $A$ and $B$ are subsets of $R$ then $(A,B)$ or $\langle A,B\rangle$ denotes the smallest ideal of $R$ containing both $A$ and $B$. If $I$ and $J$ are two ideals of $R$, then $I+J$ denotes the set $\{ i + j : i\in I, j\in J\}$. It is an easy theorem that $I+J=\langle I,J\rangle$ and so $\langle A,B\rangle = \langle A \rangle + \langle B\rangle$.