For four wide classes of topological rings $\mathfrak R$, we show that all flat left R$-contramodules have projective covers if and only are descending chains cyclic discrete right R$-modules terminate the quotient R$ perfect. Three for which this holds complete, separated associative with a base neighborhoods zero formed by open two-sided ideals such either ring is commutative, or it has count...

A module $M$ is said to be coretractable if there exists a nonzero homomorphism of every nonzero factor of $M$ into $M$. We prove that all right (left) modules over a ring are coretractable if and only if the ring is Morita equivalent to a finite product of local right and left perfect rings.