On study of the exact sequence of 𝒰-extension modules

Mahatma and Muchtadi-Alamsyah (2017) introduced the notion of the so-called 𝒰-extension modules which is the generalization of the concept of the extension modules. As the classic extension modules induce a long exact sequence of homologies from a given short exact sequence, there is an analogous result in the 𝒰 version in the form of an exact sequence consists of 𝒰-extension modules. It is just that such sequence has only limited number of nonzero modules while the classic one can contain infinitely many. Mahatma and Hadi (2020) proposed sufficient conditions for the sequence so it can be extended. However, it requires some tight assumptions to be hold. In this paper, we shall give another method of extending the sequence so we may avoid the dependency on the additional assumptions.