  Let $R$ be a commutative Noetherian ring. We give criteria for flatness of
$R$-modules in terms of associated primes and torsion-freeness of certain
tensor products. This allows us to develop a criterion for regularity if $R$
has characteristic $p$, or more generally if it has a locally contracting
endomorphism. Dualizing, we give criteria for injectivity of $R$-modules in
terms of coassociated primes and (h-)divisibility of certain $\Hom$-modules.
Along the way, we develop tools to achieve such a dual result. These include a
careful analysis of the notions of divisibility and h-divisibility (including a
localization result), a theorem on coassociated primes across a $\Hom$-module
base change, and a local criterion for injectivity.
