TITLE: Ideal Generated by two elements (Notation question)
QUESTION [23 upvotes]: given two elements $r$,$s$ in a ring $R$, are the following two notations equivalent?

$(r,s)$
$(r)+(s)$

For example, in the ring $\mathbb{Z}[X]$, is $(2,X)=(2)+(X)$?
Thanks a lot.

REPLY [15 votes]: 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 $\langle A,B\rangle = \langle A \rangle + \langle B\rangle$.