Does anyone know examples of rings $R$ with the property that any submodule of an injective (right) $R$-module is flat? If I'm not missing something, this class of rings includes the (Von Neumann) regular rings and is included into the class of (right)$IF$-rings (rings such that any injective module is flat). I'm looking for examples which are not regular. References with many examples are welcome.

absolutely flat. For some infos see mathreference.com/mod-hom-te,absflat.html – Martin Brandenburg Feb 29 '12 at 14:44