Abstract
The soft set and its direct on a ring structure is the main target of the proposed paper. The concept herein, can be considered as the connecting tool between the S-S-T, S-T and R-T. In the present paper, the basic properties for the proposed structure are developed and the relation between S-I-R and S-U-R are analyzed.
Abbreviations
Soft-Set-Theory
Set-Theory
Soft Set
Ring-Theory
Soft-Intersection-Ring
Soft-Union-Ring
semi-rings
Soft-Sub-Rings
Soft-Sub-Felds
Soft Sub-Modules
Soft-Groups
Soft-Intersection-Groups
Soft Union-Groups
Introduction
The tools of scientific research vary from time to time depending on the technological progress of each stage, but it can be said that the traditional tools used in modeling, logic, and computing are all clear, inevitable and accurate in character. However, there are many complex problems in many areas such as economic, engineering, sciences, social, and medical, all of which involve data not all of them are always fragile [1]. The main concepts and notations of the so called S-T was presented firstly in 1999 by Molodtsov [2] as a mathematical tools for treating the un-certainties. The S-T had taken attention in the structure of algebra like groups [3], S-R [4], rings [5], ordered semi-groups [6], and near-rings [7]. Furthermore, Xiao et al. [8] developed the notations of the exclusive disjunctive soft-sets and solved some of their operations. The bijective soft-set had been studied and solved some applications by Gong et al. [9]. The concepts of S-S-R and ideals of a ring, S-S-F of a field and S-S-M of a module, all were studied by AtagünandSezgin [10]. The new S-G, S-I-G and S-U-G were developed and studied by C. Agman et al., [11]. The subject of Algebraic structures of S-S had studied extensively by different authors, such as Maji et al. [12], where they developed some basic definitions on the S-S based on the analysis of several operations on S-S, see also, Ali et al. [13, 14]. In the present paper, the basic properties for the proposed structure are developed and the relation between soft intersection ring and soft-union-ring are analyzed.
Some basic information and definitions
In the present section, we will restore some basic information, and definition that will be helpful for the present paper.
Let us make the following unifying of symbols in the remaining whole text:
UCU: Common-Universe
USoft: Soft-set
UitUniv: Initial Universe
EParmeter: Set of parameters
Ppower (UitUniv): Power set
Definition (1)
Assume a soft-set takes the form ςA over UitUniv, mathematically defined as:
The soft-set USoft can be modeled as a serious number of ordered pairs, as follows:
From (2), one can notice the following: The soft-set a parameter family consists of sub-sets of USoft, The sets ςA can be arbitrary, Some of ςA may be empty sets, Some of ςA may be non-empty. Suppose that we have more soft-sets, all in sub-set A, B, C, ... of EParmeter then the S-S will take the form ςA, ςB, and ςC
Assume that ςA, and ςB, be S-S over USoft, then the
Assume that ςA, and ςB, be S-S over USoft, then the
Assume that ςA, and ςB, be S-S over USoft, then the
Assume that ςA, and ςB, be S-S over USoft, then the
Assume that ςA, and ςB, be S-S over UCU, and ζ be a function between A and B, then the
Theorem: Relative soft-set
Assume that ςA, and ςB, be S-S over UCU,
Assume that ϖ be a ring andϖ
R
be S-S over USoft, then ϖ
R
is S - I - R over USoft iff:
Herein, we will define the S-U-R, then defining S-S-U-R, S-U–ideal of a ring. The basic properties for S-U-R, S-S-U-R, and S-U–ideal w.r.t. S-S-operations will be discussed, let us now denote ϖ by any ring, with zero element 0 ϖ in the remaining of the text.
Definition (8)
Assume that (ϖ, +, .) be a ring andς ϖ be s S-S over UCU, if ς ϖ is S-U-G over UCU for the binary operation ‘+’, and ς ϖ S-U-G over UCU for the binary operation ‘.’.
Theorem
Assume that ϖ be a ring and ς
ϖ
be S-S over UCU, then ς
ϖ
will be a S-U-R iff:
Assume that ς
ϖ
be a S-U-R, then:
Therefore
On the opposite side,
Assume that
And
By taking 0 ϖ , then:
By similar procedure,
Therefore
Assume that ς
ϖ
be a S-U-R over UCU, and if
Then
Let us assume
Then
By a similar procedure,
From equations (23) and (24):
Assume that ς ϖ is a S-S on UCU, if ς ϖ (0 ϖ ) ⊆ 1 ϖ = ς ϖ (α) ∀ 0 ≠ α ∈ ϖ, then ς ϖ is a S-U-R over UCU
Let
It is required now to show:
And
To continue the proof, let α, β ∈ ϖ, we will discuss three different cases as follow:
Assume that α ≠ 0 & β = 0 or α = 0 & β ≠ 0, this leads to:
Since
Assume that α ≠ 0 & β ≠ 0, this leads to:
And
Assume that α = 0 & β = 0, this leads to:
The S-S and its direct on a ring structure was the main target of the proposed paper, here in throughout the work, we developed a new form of ring on a S-S, called S-U-R by the help of S-S and U-operation of sets. This new developed form can be viewed as the connection between the S-S-T, S-T and R-T, and so is helpful for obtaining results in the mean of ring structure. Throughout the paper, and by making use of definitions, the concepts S-S-U-R and S-U-ideals of a ring have been investigated w.r.t. S-image, and S-anti image. Last, we had obtained a meaningful relationship between S-U-R and S-I-R.