The main goal of this paper is to characterize rings over which the
mininjective modules are injective, so that the classes of mininjective modules
and injective modules coincide. We show that these rings are precisely those
Noetherian rings for which every min-flat module is projective and we study
this characterization in the cases when the ring is Kasch, commutative and when
it is quasi-Frobenius. We also treat the case of $n\times n$ upper triangular
matrix rings, proving that their mininjective modules are injective if and only
if $n=2$.
We use the developed machinery to find a new type of examples of indigent
modules (those whose subinjectivity domain contains only the injective
modules), whose existence is known, so far, only in some rather restricted
situations.

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