Let $R$ be a ring with unit. A submodule $N$ of an $R$-module $M$ is called superfluous if the only sumbodule $T$ of $M$ for which $N+T = M$ is $M$ itself.

It is shown, for example, in

[1] F. W._Anderson, K. R. Fuller "Rings and Categories of Modules" (1974)

that if every submodule of $M$ is contained in a maximal submodule, then the radical of $M$ is superfluous (Proposition 9.18). This, in particular, implies that for every finitely generated module $M$ its radical is superfluous. In exercise 9.2. it is explained that divisible abelian groups coincide with their radicals, and therefore their radicals are not superfluous. Divisible abelian groups are not projective objects.

I was curious if it is possible to construct a projective module with non-superfluous radical.

Question: is there an example of a ring $R$ and a projective $R$-module $P$ such that the radical $JP$ of $P$ is not superfluous?

The existence of such modules (or, at least, that its non-existence is non-obvious) is somehow hinted by the formulation of Corollary 17.12 in [1]:

Let $J = J(R)$. If $P$ is a projective left $R$-module such that $JP$ is superfluous in $P$ (e.g., if ${}_RP$ is finitely generated), then $J(End({}_RP)) = Hom_R(P,JP)$ and $End({}_RP)/J(End_RP) \cong End({}_RP/JP)$.