This paper studies two dual notions in module theory—namely, radicals and socles—from the standpoint of reverse mathematics. We first consider radicals of $\mathbb{Z}$-modules, where the radical of a $\mathbb{Z}$-module $M$ is defined as the intersection of $pM=\{px:x\in M\}$ with $p$ taken from all primes. It shows that ${\mathrm{ACA}}_0$ is equivalent to the existence of radicals of $\mathbb{Z}$-modules over ${\mathrm{RCA}}_0$. We then study socles of modules over commutative rings with identity. The socle of an $R$-module $M$ is the largest semisimple submodule of $M$. We show that the existence of socles of modules over a commutative ring with identity is equivalent to ${\mathrm{ACA}}_0$ over ${\mathrm{RCA}}_0$. Vector spaces are semisimple modules over fields. In general, semisimple modules possess nice properties of vector spaces. Lastly, we study characterizations of semisimple modules over commutative rings using techniques of reverse mathematics.