WEAK $({C_{11}}^+)$ MODULES WITH ACC OR DCC ON ESSENTIAL SUBMODULES

Abstract

In this note we study modules with $(W{C_{11}}^{+})$ property. We prove thatif $M$ satisfies $(W{C_{11}}^{+})$ and $M/(Soc M)$ has finite uniform dimensionthen $M=M_{1}\oplus M_{2}$ where $M_1$ is semisimple and $M_2$ with finiteuniform dimension. In particular, if $M$ satisfies $(W{C_{11}}^{+})$ andascending chain (respectively, descending chain) condition on essentialsubmodules then $M=M_{1}\oplus M_2$ for some semisimple submodule $M_1$and Noetherian (respectively, Artinian) submodule $M_2$.