Abstract
For four wide classes of topological rings
R
\mathfrak{R}
, we show that all flat left
R
\mathfrak{R}
-contramodules have projective covers if and only if all flat left
R
\mathfrak{R}
-contramodules are projective if and only if all left
R
\mathfrak{R}
-contramodules have projective covers if and only if all descending chains of cyclic discrete right
R
\mathfrak{R}
-modules terminate if and only if all the discrete quotient rings of
R
\mathfrak{R}
are left perfect.
Three classes of topological rings for which this holds are the complete, separated topological associative rings with a base of neighborhoods of zero formed by open two-sided ideals such that either the ring is commutative, or it has a countable base of neighborhoods of zero, or it has only a finite number of semisimple discrete quotient rings.
The fourth class consists of all the topological rings with a base of neighborhoods of zero formed by open right ideals which have a closed two-sided ideal with certain properties such that the quotient ring is a topological product of rings from the previous three classes.
The key technique on which the proofs are based is the contramodule Nakayama lemma for topologically T-nilpotent ideals.