ON G-RADICAL SUPPLEMENT SUBMODULES

In this work, we give some new properties of Rad-supplement and g-radical supplement submodules. Let V be a g-radical supplement of U in M and U or V be essential submodule of M. Then Rad(g)V = V boolean AND Rad(g)M. Let V be a g-radical supplement of U in M, U or V be essential submodule of M and x is an element of V. Then Rx <<(g) V if and only if Rx <<(g) M. In this work, some relations between Rad-supplement, g-radical supplement, beta* and beta(g)* relations are also studied. Let X beta(g)*Y in M. If V is a g-radical supplement of X in M and V (sic) M, then V is also a g-radical supplement of Y in M. Let M be an R-module. It is proved that M is semilocal (g-semilocal) if every submodule of M beta* equivalent to a Rad-supplement (g-radical supplement) submodule in M.