Questions regarding rings where every injective modules over them are also flat modules. Let $R$ be a ring such that every left injective $M$ module over $R$ is also a left flat module over $R$. An example that springs to my mind to illustrate this type of rings are the regular rings, since it is well known that every module over a regular ring is flat, particularly all injective modules over a regular ring are flat.
First question is: Are there any more examples of these type of rings, where every injective module over this particular ring is also a flat module over the same ring?
Second question is if $R$ is ring such that every left injective $M$ module over $R$ is also a left flat module over $R$ and every right injective $M$ module over $R$ is also a right flat module over $R$, then is it possible to prove that $R$ is a coherent ring? That is, where every submodule of a finitely generated module over $R$ is finitely presented.
I've run out of ideas proving this last statement but maybe my affirmation is not true.
 A: A ring for which injective right modules are flat are called right IF rings, and they've been studied before. Take a look at

Jain, Saroj. "Flat and FP-injectivity." Proceedings of the American Mathematical Society 41.2 (1973): 437-442.

Coherence and the IF condition are linked there with a third condition, FP-injectivity.
It looks like there are a couple of results you would be interested in, especially Prop 3.8: if $R_R$ is a cogenerator in the category of right $R$ modules, then $R$ is right IF iff it is left coherent.
The paper ends with the remark "We do not know if right IF rings are left coherent, even for commutative rings."
I think I would start looking to see if any subsequent citations of this paper undertook that problem.
As for your first question "are there nonregular rings where injectives are flat?" there is, of course, the obvious case of quasi-Frobenius rings, whose injective right modules are precisely the projective right modules, hence they are all flat too.  This condition is "orthogonal" to regularity in the sense that a quasi-Frobenius regular ring is already semisimple.  The simplest nontrivial example would be, perhaps, $\mathbb Z/4\mathbb Z$.
