A nonempty subset is a submodule once it is closed under $rx+y$.Suggest a slogan
Let \(R\) be a ring and \(M\) an \(R\)-module. A subset \(N\subseteq M\) is an \(R\)-submodule if and only if \(N\) is nonempty and \(rx+y\in N\) for all \(r\in R\) and \(x, y \in N\).
A submodule contains \(0\) and is closed under addition and scalar multiplication, so it satisfies the condition. Conversely, choose \(x\in N\); taking \(r=-1\) and \(y=x\) gives \(0\in N\). Taking \(y=0\) gives \(rx\in N\) for all \(r\in R\) and \(x\in N\), in particular \(-x\in N\), and taking \(r=1\) gives \(x+y\in N\). Hence \(N\) is an additive subgroup of \(M\) closed under scalar multiplication.