ON H-COFINITELY SUPPLEMENTED MODULES

A module MM is called emphHemphH-cofinitely supplemented if for every cofinite submodule EE (i.e. M/EM/E is finitely generated) of MM there exists a direct summand DD of MM such that M=E+XM=E+X holds if and only if M=D+XM=D+X, for every submodule XX of MM. In this paper we study factors, direct summands and direct sums of emphHemphH-cofinitely supplemented modules. Let MM be an emphHemphH-cofinitely supplemented module and let NleqMNleqM be a submodule. Suppose that for every direct summand KK of MM, (N+K)/N(N+K)/N lies above a direct summand of M/NM/N. Then M/NM/N is emphHemphH-cofinitely supplemented. Let MM be an emphHemphH-cofinitely supplemented module. Let NN be a direct summand of MM. Suppose that for every direct summand KK of MM with M=N+KM=N+K, NcapKNcapK is also a direct summand of MM. Then NN is emphHemphH-cofinitely supplemented. Let M=M1oplusM2M=M1oplusM2. If M1M1 is radical M2M2-projective (or M2M2 is radical M1M1-projective) and M1M1 and M2M2 are emphHemphH-cofinitely supplemented, then MM is emphHemphH-cofinitely supplemented