Decomposing modules into direct sums of submodules with types Original Research Article

In this paper, using type submodules, we introduce TS-modules and TS-modules with (T3). These are weaker concepts than those of extending modules and quasi-continuous modules. We show that the entire decomposition theory of extending modules and quasi-continuous modules in Mohamed and Müller (Continuous and Discrete Modules, Cambridge Univ. Press, Cambridge, 1990) has an analogue for TS-modules and TS-modules with (T3). Besides, the study of TS-modules and TS-modules with (T3) also brings out a new concept of rings, namely, rings satisfying t-acc.