The X[[S]]-Sub-Exact Sequence of Generalized Power Series Rings

*Correspondence: ahmadfaisol@fmipa.unila.ac.id Let R be a ring, (S, +, ≤) a strictly ordered monoid, and K, L, M are R-modules. Then, we can construct the Generalized Power Series Modules (GPSM) K[[S]], L[[S]], and M[[S]], which are the module over the Generalized Power Series Rings (GPSR) R[[S]]. In this paper, we investigate the property of X[[S]]-subexact sequence on GPSM L[[S]] over GPSR R[[S]].


Introduction
A non-empty set of with an associative binary " * " is called a semigroup. If has an identity element, then ( , * ) is called a monoid. Furthermore, if each element of has an inverse, then ( , * ) is called a group (Howie 1995). A ring ( , +,•) is a non-empty set of R with two binary operations. (R, +) is a commutative group, (R,•) a semigroup, and satisfies the left and right distributive laws (Adkins and Weintraub 1992).
One example of a ring is the polynomial ring [ ], which is defined as the set of all functions from non-negative integers ℕ⋃{0} to ring R with finite support. Furthermore, this ring is generalized into the power series ring [ ] by removing the finite support conditions (Hungerford 1974). Furthermore, the polynomial ring [ ] can be generalized by changing its function domain to any semigroup. This ring is then known as the semigroup ring and is denoted by [ ] (Gilmer 1984).
It is known that a ring can be seen as a module over itself.  (Faisol, Surodjo, and Wahyuni 2019c), and applies the relationship between almost generated module, almost Noetherian module and -Noetherian module (Faisol, Surodjo, and Wahyuni 2019b).
The Noetherian properties of an -module can be investigated through an exact sequence. If there is an exact sequence → → where and are Noetherian, then is a Noetherian -module (Wisbauer 1991). The generalization of the exact sequence in themodule is investigated by (Davvaz and Parnian-Garamaleky 1999). This result is obtained by replacing submodule 0 with submodule U ⊆ C, called the U-exact sequence. Another study related to the properties of the U-exact sequence can be seen in ((Davvaz and Shabani-Solt 2002) (Anvariyeh and Davvaz 2005)). Motivated by the U-exact sequence definition, the X-sub-exact sequence concept was introduced in (Fitriani, Surodjo, and Wijayanti 2016), which is a generalization of the exact sequence. Besides that, the generalization of an R-module generator to become a U-generator has been reviewed in (Fitriani, Wijayanti, and Surodjo 2018b). Furthermore, by using the concept of sub-linearly independent modules (Fitriani, Surodjo, and Wijayanti 2017), a basis and free module relative to a family of modules over R can be defined (Fitriani, Wijayanti, and Surodjo 2018a ]. Besides, this also provides an opportunity to investigate the properties that satisfy them.

The Research Methods
The research methods are based on the study of literature. They relate to the concept of partially ordered set, strictly ordered monoid, Artinian and narrow properties, generalized power series rings (GPSR), generalized power series modules (GPSM), exact-sequences, and X-subexact sequences. The results of this study obtained by constructing the exact sequence and X[[S]]-sub-exact sequence over an R[[S]]-module, as well as investigating the properties that apply in it.

The Results of the Research and the Discussion
Before discussing the definition and properties of the X
We were given a strictly ordered monoid ( , +, ≤) and commutative ring R with unit element 1. Next, is defined as the set

supp( ), ∈ [[S]], and , ∈ [[S]], it can be shown that M[[S]] is an R[[S]]-module. This module is called the Generalized Power Series Module (GPSM).
The following is the definition of the exact sequence and X-sub-exact sequence over an Rmodules. Let R be a ring and an R-module for each i. R-module sequence … → −1 → +1 → +1 → ⋯ is said to be exact in if there are R-homomorphism and +1 that satisfies ( ) = Ker( +1 ). The sequence is said to be exact if it is exact at every . Furthermore, this exact sequence is generalized to the X-sub-exact sequence. Suppose K, L, M are modules over R and X is a submodule of L. Triple (K, L, M) is said to be X-sub-exact over L if there are R-homomorphism f and g such that the sequence → → is the exact sequence over R-modules.