Example 87.4.1. Let R be a linearly topologized ring and let M be a linearly topologized R-module. Let I_\lambda be a fundamental system of open ideals in R and let M_\mu be a fundamental system of open submodules of M. The continuity of + : M \times M \to M is automatic and the continuity of R \times M \to M signifies
Since fM_\nu + I_\lambda M_\nu \subset M_\mu if M_\nu \subset M_\mu we see that the condition is equivalent to
However, it need not be the case that given \mu there is a \lambda such that I_\lambda M \subset M_\mu . For example, consider R = k[[t]] with the t-adic topology and M = \bigoplus _{n \in \mathbf{N}} R with fundamental system of open submodules given by
Since every x \in M has finitely many nonzero coordinates we see that, given m and x there exists a k such that t^ k x \in M_ m. Thus M is a linearly topologized R-module, but it isn't true that given m there is a k such that t^ kM \subset M_ m. On the other hand, if R \to S is a continuous map of linearly topologized rings, then the corresponding statement does hold, i.e., for every open ideal J \subset S there exists an open ideal I \subset R such that IS \subset J (as the reader can easily deduce from continuity of the map R \to S).
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