Definition 11

The annihilator of a subset S of an A-module E is the set of elements $\alpha \in A$ such that $\alpha x = 0$ for all $x \in S$.

The annihilator of S is usually denoted by Ann(S); for a subset S consisting of a single element x, we write Ann(x) instead of Ann(\{x\}) and call Ann(x) the annihilator of x.

The relation $\alpha x = 0$ may also be expressed by saying that x is annihilated by $\alpha$.

It is immediate that the annihilator of an arbitrary subset S of E is a left ideal of A; for it to be equal to A, it is necessary and sufficient (by virtue of (M_IV)) that $S = \{0\}$. If two subsets S, T of E are such that $S \subset T$, the annihilator of T is contained in the annihilator of S. If $(S_i)_{i \in I}$ is an arbitrary family of subsets of E, the annihilator of the union $\bigcup_i S_i$ is the intersection of the annihilators of the $S_i$. In particular, the annihilator of a subset S of E is the intersection of the annihilators of the elements of S. To say that an element of E is free is equivalent to saying that its annihilator is $\{0\}$. For all $x \in E$ and all $\alpha \in A$, the annihilator of $\alpha x$ is the set of $\beta \in A$ such that $\beta \alpha \in \mathrm{Ann}(x)$.

The annihilator of a submodule M of E is a two-sided ideal of A; for, if $\alpha x = 0$ for all $x \in M$, then also $\alpha (\beta x) = 0$ for all $x \in M$ and all $\beta \in A$, hence $\alpha \beta$ belongs to the annihilator of M for all $\beta \in A$. In particular, the annihilator of E is a two-sided ideal of A.

For all $\alpha \in A$, let $h_\alpha$ be the homothety $x \mapsto \alpha x$; it is known that the mapping $\alpha \mapsto h_\alpha$ of A into the endomorphism ring $\mathscr{E} = \mathrm{Hom}_\mathbf{Z}(E, E)$ of the commutative group (without operators) E, is a ring homomorphism (§ 2, no. 5). The inverse image of 0 under this homomorphism is the annihilator $a$ of E; the image of A under the homomorphism $\alpha \mapsto h_\alpha$ is therefore isomorphic to the quotient ring $A/a$. The module E is called faithful if its annihilator $a$ reduces to 0.

Let E be any A-module, $a$ a two-sided ideal of A contained in Ann(E) and let $\dot{\alpha}$ be an element of the quotient ring $A/a$; for all $x \in E$, the element $\alpha x$ is the same for all the $\alpha \in A$ belonging to the class $\dot{\alpha}$ mod. $a$; if this element is denoted by $\dot{\alpha} x$, it is immediately seen that the mapping (α, x) ↦ αx defines (with addition on E) an (A/α)-module structure on E. When α = Ann(E), the (A/α)-module E thus defined is faithful; we shall say that it is the faithful module associated with the A-module E. Observe that every submodule of an A-module E is also a submodule of the associated faithful module and conversely.