TOPOLOGICAL MODULES OVER STRICTLY MINIMAL TOPOLOGICAL RINGS

Dinamérico P. Pombo Jr., Instituto de Matemática, Universidade Federal Fluminense, Rua Mário Santos Braga, s/n, 24020-140, Niterói-RJ, Brasil.

Throughout all modules are unitary left R-modules, where $(R , {\tau}_R )$ is a separated topological ring with a non-zero identity element. $(R , {\tau}_R )$ is strictly minimal if, for every separated topology $\theta$ on R such that $(R , \theta)$ is a topological $(R , {\tau}_R )$-module (R endowed with its canonical left R-module structure), we have that $\theta = {\tau}_R$.

Theorem 1. The following conditions are equivalent:

(a) $(R , {\tau}_R )$ is strictly minimal;

(b) for every separated topological $(R , {\tau}_R )$-module F which is a free R-module of dimension 1, every R-module isomorphism from R into F is a homeomorphism;

(c) for every free R-module F of dimension 1, there is only one separated $(R , {\tau}_R )$-module topology on F;

(d) for every topological $(R , {\tau}_R )$-module E and for every separated topological $(R , {\tau}_R )$-module F which is a free R-module of dimension 1, every surjective R-linear mapping from E into F with a closed kernel is continuous;

(e) for every topological $(R , {\tau}_R )$-module E and for every separated topological $(R , {\tau}_R )$-module F which is a free R-module of dimension 1, every R-linear mapping from E into F with a closed graph is continuous.

Corollary. Assume that $(R , {\tau}_R )$ is strictly minimal. If E is a topological $(R , {\tau}_R )$-module and F is a separated topological $(R , {\tau}_R )$-module which is a free R-module of dimension 1, then every surjective continuous R-linear mapping from E into F is open.

Theorem 2. Assume that $(R , {\tau}_R )$ is strictly minimal and complete. Then, for every $n\in {\mathbb N}^{\ast}$ and for every separated topological $(R , {\tau}_R )$-module F which is a free R-module of dimension n, every R-module isomorphism from Rⁿ into F is a homeomorphism.

Corollary 1. Let $(R , {\tau}_R )$ be as in Theorem 2. Then: (a) for every $n\in {\mathbb N}^{\ast}$ and for every free R-module of dimension n, there is only one separated $(R , {\tau}_R )$-module topology on F; (b) if E is a separated topological $(R , {\tau}_R )$-module and M is a submodule of E which is a free R-module of dimension $n\in {\mathbb N}^{\ast}$, then M is closed.

Corollary 2. Assume that R is a principal ring and that $(R , {\tau}_R )$ is strictly minimal and complete. Let $n\in {\mathbb N}^{\ast}$, E a separated topological $(R , {\tau}_R )$-module which is a free R-module of dimension n and F a separated topological $(R , {\tau}_R )$-module which is a free R-module. Then every R-linear mapping from E into F is continuous.