Definition 1

Let A be a ring. The radical of an A-module M is the submodule defined as the intersection of the maximal submodules of M (VIII, p. 48, Definition 2) or, equivalently, the set of elements of M annihilated by every homomorphism from M to a simple A-module.

In the remainder of this chapter, we denote the radical of an A-module M by $\mathfrak{R}_A(M)$ or simply $\mathfrak{R}(M)$.

Let A be a ring. The radical of an A-module M is reduced to 0 (in which case we say, by abuse of language, that M is without radical) if and only if there exist a family $(S_i)_{i\in I}$ of simple A-modules and a family $(f_i)_{i\in I}$ of A-linear mappings $f_i: M\rightarrow S_i$ such that we have $\cap_{i\in I}$ Ker($f_i$) $= 0$. This is equivalent to M being isomorphic to a submodule of a product of simple A-modules.