Some exact sequence of ideals and quotients

The maps are module homomorphisms; more precisely, they are:

$I \cap J \mapsto I \oplus J $ via $x \mapsto (x,x)$,

and

$I\oplus J \mapsto I + J $ via $(x,y) \mapsto x - y$.

If you embed this exact sequence as a subsequence of the obvious short exact sequence

$ 0 \to R \to R\oplus R \to R \to 0$

(with maps defined by the same formulas), then the quotient is the second short exact sequence that you ask about.

Matt E. answers the question, but I just wanted to add a couple of things.

First note a proper ideals of $R$ does not inherit a ring structure from $R$ because it does not contain $1$. Also, there is no notion of exact sequences for rings, since the kernel of a ring homomorphism is not a ring (but an ideal).

The maps are $R$-module homomorphisms, so indeed the first and last term of the first sequence are ideals of $R$. The middle term $I \oplus J$ is just the direct sum of $I$ and $J$ as $R$-modules; it is not an ideal of $R$.

The second exact sequence comes from applying the snake lemma to the morphism of short exact sequences which Matt points to; its "cokernel" part is your sequence.