Summary: The aim of this paper is to investigate graded almost prime submodules under localizations by introducing a new definition, their characterizations and their properties. Let \(G\) be an abelian group with identity \(e\), \(R\) be a \(G\)-graded commutative ring and \(M\) a graded \(R\)-module.
A proper graded submodule \(N\) of \(M\) is a graded almost prime if for \(r\in h(R)\) and \(m\in h(M)\), \(rm\in N - (N : M)N\) implies \(r\in (N : M)\) or \(m\in N\). A proper graded submodule is a graded weakly prime submodule if for \(r\in h(R)\) and \(m\in h(M)\), \(0 = rm \in N\) implies \(r\in (N : M )\) or \(m\in N\). Every graded prime submodule is graded weakly prime as well as graded almost prime. However, since \(\{0\}\) is always a graded weakly prime submodule and hence a graded almost prime submodule. A graded almost prime submodule need not be a graded prime submodule.