  Let $E$ be an arbitrary (countable) graph and let $R$ be a unital commutative
ring. We analyze the ideal structure of the Leavitt path algebra $\lr$
introduced by Mark Tomforde. We first modify the definition of basic ideals and
we then develop the ideal characterization of Mark Tomforde. We also give
necessary and sufficient conditions for the primeness and the primitivity of
$\lr$. Then by applying these results we determine prime graded basic ideals
and left (or right) primitive graded ideals of $\lr$. In particular, we show
that when $E$ satisfies Condition (K) and $R$ is a field, the set of prime
ideals and the set of primitive ideals of $\lr$ coincide.
