A submodule N of an R-module M is lifting if \(\text {Ann}_R(\frac{M}{N})\) is a lifting ideal of R, i.e., if \(r(1-r)M\subseteq N\) for some \(r\in R\) , then there exists an idempotent \(e\in R\) such that \((r-e)M\subseteq N\) . We show that R is a clean ring if and only if every submodule of any R-module M is lifting. A lifting comaximal decomposition of a submodule N of an R-module M is \(N=\displaystyle \bigcap _{i=1}^{n}N_i\) , where \(N_i's\) are lifting submodules of M such that \((N_i:M)+(N_j:M)=R\) , for all \(i\ne j\) . If R is a Noetherian ring and M is a finitely generated multiplication R-module, then we show that every proper submodule of M has a unique (up to order) lifting comaximal decomposition. We will also examine lifting submodules from the point of view of the Zariski topology.
