Cryptography over quasi-fields (which are not field, but where non-invertible elements are hard to find) is very common. This includes many cryptosystems such as RSA, but also Rabin, Goldwasser-Micali, Benaloh, Okamoto-Uchiyama, Naccache-Stern, Paillier, Damgard-Jurik, BCP, and many other related cryptosystems. This also includes all works based on composite-order elliptic curves, such as the BGN cryptosystem.
These quasi-fields, that we usually call RSA groups, have nice properties that are not easily found in prime order field. This allows for the construction of cryptosystems with very cool properties: Cock's cryptosystem is a simple identity-based encryption scheme, BCP (mentioned above) has a double trapdoor mechanism (many public key / private key pairs can be generated, and a global master secret key can decrypt them all), Paillier is additively homomorphic, with efficient decryption, etc.
Composite-order rings also have the advantage of containing groups of unknown order. For example, the subgroup of squares over an RSA group of order $n = pq$ has order $\varphi(n)/4 = (p-1)(q-1)/4$, which is unknown as knowing it is equivalent to knowing the factorisation. This allows for example to build constant size range proofs (zero-knowledge arguments for membership to an interval).
The phi-hiding assumption over composite order groups also gives rise to cryptographic schemes with efficiency properties that are very hard to obtain otherwise, such as private information retrieval schemes.
In elliptic curve cryptography, it is now very common to first design schemes over composite-order elliptic curves, which are far easier to manipulate because they contain "hidden subgroups" in which all secret informations are placed (these groups can only be "accessed" given the factorization), before trying to reproduce these results on prime order groups - and for several constructions, we still don't know if using prime order groups is possible.