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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.

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I am confused: any module is a submodule of an injective module (its injective hull). So, you are basically asking for rings on which every module is flat? –  Liran Shaul Feb 29 '12 at 9:29
    
Based on Liran's observation: Since projective modules are flat (if the ring has a unit), the semisimple Artinian rings satisfy your requirement. Compare mathoverflow.net/questions/62464/…. –  Ralph Feb 29 '12 at 10:05
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A ring over which every module is flat is called absolutely flat. For some infos see mathreference.com/mod-hom-te,absflat.html –  Martin Brandenburg Feb 29 '12 at 14:44
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You're right that von Neumann regular rings satisfy this condition: these are precisely the rings over which every right (or equivalently, left) module is flat. (For instance, see Lam's textbook Lectures on Modules and Rings, section 4B.) Thanks to Liran Shaul's comment, we see that these are the only examples of rings satisfying the condition that you want. –  Manny Reyes Feb 29 '12 at 16:38
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Thanks for your answers. As Liran Shaul noticed I was actually asking for rings on which every module is flat:), thus only the Von Neumann regular rings! –  todea Mar 2 '12 at 21:52
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