Definition 3

Given a topological ring $A$ with an identity element, a topological left $A$*-module is a set* $E$, together with:

1) a left $A$*-module structure*; 2) a topology compatible with the additive group structure of $E$, and satisfying the following axiom: (MT) The mapping $(\lambda, x) \to \lambda x$ of $A \times E$ into $E$ is continuous.

We define similarly the notion of a topological right $A$-module; since every right $A$-module can be considered as a left $A^0$-module, where $A^0$ is the opposite ring of $A$, and since the topology of $A$ is compatible with the ring structure of $A^0$, there is no need to distinguish between topological right $A$-modules and topological left $A^0$-modules.