Some Studies Of Socle And Radical Of Semimodules

In this dissertation we mainly investigate the scole of well-semimodules andscole、weak-radical、radical of semimodules in the category of well-semimodulesand semimodules.The pseudo-semisimple decomposition of well-semimodules andthe function of radical (weak–socle) over finitely generated(cogenerated)semmimodules are studied. The dissertation falls into four parts.In the first section, we study the association of semisimple decomposition andpseudo-semisimple decomposition.Then we generalize semisimple decomposition ofmodules to pseudo-semisimple decomposition of well-semimodules.In the second section,we use the trace of paper [12] to definite the socle ofwell-semimodules and obtain any well-semimodules has socle.We also study theproperty of well-semimodules and definite the weak-socle of semimodules and obtainSocM WSocM,in trun,is not established,in addition,in division semiring.Similarythe studying of socle,we respectively use reject and the intersection of maximalsubsemimodules to definite the weak-radical and radical of semimomdules anddisscuss their prosperities.In the third section,we generalize the prosperities and results of finitelygenerated(cogenerated) modules to the category of semimodules.In the last,we study the function of weak-socle and radical over finitelygenerated(cogenerated) semimodulesin the category of well-semimodules and obtainthe necessary condition for semimodule M is finitely generated and the sufficientand necessary condition for semimodule M is finitely cogenerated.