Variants of absolute direct summand property.

A right module M over a unital ring R is said to be absolute direct summand (for short ADS) providedM = S⊕T ′ for every submodules S, T, T ′ such thatM = S⊕T and T ′ is a complement of S. However the concept was introduced already by Laslo Fuchs in late 60’s, only recent results show importance of the notion for research of relative injectivity in module categories and for structure of corresponding rings [1]. The natural generalization of the notion provided by restriction on a choice of T ′ in the definition to some particular classes of submodules appears to be a useful tool for further study of injectivity properties of modules and rings. In particular, M is called an essentially ADS-module if M = S ⊕ T ′ for each decomposition M = S ⊕ T and each complement T ′ of S with T ′ ∩ T = 0 and S ∩ (T ′ ⊕ T ) ≤ S, and M is type ADS if M = S ⊕ T ′ for each decomposition M = S ⊕ T with type submodules S, T and each type complement T ′ of S. We will discuss possible generalization of the following basic structural results about variants of ADS-modules: