ON FS-LIFTING AND CFS-LIFTING SEMIMODULES OVER SEMIRINGS

In this paper, we introduce the notions of fs-lifting and cfs-lifting semimodules as respectively the generalizations of finitely lifting (f-lifting for short) semimodules and cofinitely lifting (cf-lifting) semimodules. Under some conditions, we prove some results on fs-lifting and cfs-lifting semimodules for proving the equivalence between fs-lifting and cfs-lifting semimodules. An $R$-semimodule $M$ is fs-lifting if every finitely generated subtractive subsemimodule $N$ of $M$, there exists a direct summand $K$ of $M$ such that $K\leq N$ and $N/K\ll M/K$; so if every coessential finitely generated subtractive subsemimodule $N$ of $M$, there exists a direct summand $K$ of $M$ such that $K\leq N$ and $N/K\ll M/K$, $M$ is called cfs-lifting.