Corollary 1

Let M be an A-module and N a submodule of M.

a) We have $\mathfrak{R}(N)\subset \mathfrak{R}(M)\cap N$.

b) We have $(\mathfrak{R}(M) + N)/N\subset \mathfrak{R}(M/N)$. If N is contained in $\mathfrak{R}(M)$, then we have the equality $\mathfrak{R}(M/N) =\mathfrak{R}(M)/N$.

c) The module $M/\mathfrak{R}(M)$ is without radical. If the module $M/N$ is without radical, then we have $\mathfrak{R}(M)\subset N$.

Assertion a) follows from Proposition 1 applied to the canonical injection from N to M, and assertion b) follows from Proposition 1 applied to the canonical mapping from M to $M/N$. From b), we deduce that $M/N$ is without radical if $N =\mathfrak{R}(M)$ and that we have $\mathfrak{R}(M)\subset N$ if $M/N$ is without radical.

It follows from Example 1 that there can exist submodules N containing $\mathfrak{R}(M)$ such that the radical of $M/N$ is not zero.