Contramodules over pro-perfect topological rings

TL;DR

This paper characterizes when all flat contramodules over certain topological rings have projective covers, linking it to properties of discrete quotient rings and introducing a contramodule Nakayama lemma for topologically T-nilpotent ideals.

Contribution

It identifies specific classes of topological rings where flat contramodules have projective covers, extending the understanding of contramodule theory in topological algebra.

Findings

All flat contramodules have projective covers iff all are projective.

Termination of descending chains of cyclic discrete modules characterizes the rings.

The contramodule Nakayama lemma is a key tool in the proofs.

Abstract

For four wide classes of topological rings $\mathfrak R$, we show that all flat left $\mathfrak R$-contramodules have projective covers if and only if all flat left $\mathfrak R$-contramodules are projective if and only if all left $\mathfrak R$-contramodules have projective covers if and only if all descending chains of cyclic discrete right $\mathfrak R$-modules terminate if and only if all the discrete quotient rings of $\mathfrak R$ 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.