Some exact sequence of ideals and quotients I saw an exact sequence of ideals 
$$0 \rightarrow I \cap J\rightarrow I \oplus J \rightarrow I + J \rightarrow 0$$In this sequence, maps are ring homomorphisms or module homomorphisms?
And how the above sequence yield the exact seqeunce 
$$0 \rightarrow R/I \cap J\rightarrow R/I \oplus R/J \rightarrow R/(I + J) \rightarrow 0$$
 A: 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.
A: 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.
