The class of semi-modules with direct summands are stable

This work aims to introduce and explore the concept of SS-semimodules over semirings to generalize fully stable semimodules and duo semimodules. We call an S-semimodule C is SS-semimodule if every direct summand of C is stable. Equivalent expressions, properties, and related results of SS-semimodules are given. We prove that an S-semimodule C becomes SS-semimodule in case every decomposition C = H ⊕ K, HomS (H, K) = 0. Results regarding fully stable semimodules were generalized to SS-semimodules. For example, if C is a regular SS-semimodule, then EndS (C) is fully stable. Known semimodules related to SS-semimodules were investigated. We obtain that, every multiplication semimodule is SS-semimodule. Also,the SS-semimodule property satisfies the summand intersection property. Moreover, we discuss the direct sum property and inherited property for SS-semimodules.