Is the relationship $IJ \subseteq I \cap J \subseteq I, J \subseteq I + J$ correct?
Quick sketches:
1. $IJ \subseteq I \cap J$
Elements of $IJ$ are linear combinations of the form $\sum_k i_k j_k$. Ideals are closed under multiplication with the whole ring, thus each $i_k j_k = i_k'$ Ideals are also closed under addition. Thus, each term in $\sum_k i_k j_k = \sum_k i'_k = i'' \in I$. Similarly, we can group terms according to $J$ to arrive at $\sum_k i_k j_k = \sum_k j_k' = j'' \in J$.
2 $I \cap J \subseteq I, J$
Immediate from definition of intersection.
3. $I, J \subseteq I + J$
each $i \in I$ can be written as $(i \in I) + (0 \in J)$ which is an element of $i + j$. Similarly with $J$.
Is there some way to organise these containments into a nice exact sequence? In general, I find that I have to pause and think before I remember these relations. Is there some way to organise these better?