A second reason that a hash is usually present in RSA signature schemes (apart from being able to sign long messages) is to prevent existential forgery attacks. These look like this:
Assume we have the public key $n$, $e$. Choose some random garbage $s$ (smaller than $n$), and calculate $m = s^e \mod n$ (i.e. "RSA encryption"). If you used "text book RSA signature" (without a hash and any padding), you now have a message $m$ with a fitting signature $s$.
With a hash, the signature checking equation is $H(m) \equiv s^e \mod n$ instead of $m \equiv s^e \mod n$, and this does not allow to create a message $m$ fitting to an arbitrary signature $s$ (assuming the hash function is preimage-resistant). Beware however that it is remains possible to create working $(m,s)$ pairs given the capability to obtain the signatures $s_j$ of chosen messages $m_j\ne m$, using an attack devised by Desmedt and Odlyzko (see section 3 of this paper).
(The padding scheme normally present in signatures serves the same purpose, and is designed to also resist chosen-messages attacks.)