Faithful module

A module M on a ring has E is known as faithful if its canceler is tiny room to {0}, in other words, if the action of each $\ alpha \ in has \ setminus \ {0 \}$ is noncommonplace ( $\ alpha \ cdot X \ neq 0$ for some $x \ in M$ ). In other words, a module is faithful if the associated representation $\ psi: With \ to End (M)$ is Injective.

With each module, one can associate a faithful module while proceeding in this manner. The Morphisme of rings $\ psi: With \ to End (M)$ factorizes in an injective morphism $\ tilde \ psi: \ Ker \ psi \ to End (M) has$ . As $\ ker \ psi$ is not other than Ann (M), $\ tilde \ psi$ gives to M a structure of $A/Ann (M)$ -module, and this time M is faithful since $\ tilde \ psi$ is injective.

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