Abstract

The researcher summarizes the existing in modern sociology information about main organizational forms of religions: a sect, a cult, a denomination and a church. First of all the author differentiates theological and scientific approaches to the phenomenon of the sect. Then he concentrates on the concepts of the sect (established sect), the cult, the audience cult and the client cult, indicating their social features and spread in the modern society.

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INTRODUCTION

Definitions and notations: In mathematics, the quaternions are a real scalar linear combination and three orthogonal imajinary units (denoted by i, j and k) with real coefficients that can be written as:

Where:

For more properties of quaternions, the reader is referred to Shomake and Quaternions. In this study, it is necessary to know the definition of skew polynomial ring and the center of ring. It can be found by Amir (2016, 2017, 2018) and (Amir et al. 2017).

Definition 1.1; McConnel and Robson (1987): Let R be a oring, σ be an endomorphism for R and δ be a σ-derivation. The skew polynomial ring over R is a ring that consists of all polynomials over R with an indeterminate x denoted by:

With multiplication rule, for all, a∈R, then:

For cases where δ = 0, the notations R[x; σ, δ] can be written as R[x; σ.] Moreover, the structure of skew polynomial ring R[x; σ] is different with R[x; σ, δ].

Definition 1.2; McConnel and Robson (1987): Let, R be a ring, the center of R denoted by Z(R) is defined as:

THE MAIN RESULTS

In this part, we give the forms of center of skew polynomial ring over a couple of quaternions. Let:

Where:

It is easy to show that σ is an endomorphism on H×H. So, we have H×H [x; σ] is a skew polynomial ring. Center of ring H×H [x; σ] is stated in the following theorem:

Theorem 2.1:

Proof: We show that:

Where:

The proof will be divided into two parts, i.e., P⊆Z (H×H [x; σ]) and Z(H×H [x; σ])⊆P. First, we show that P⊆Z (H×H [x; σ]):

We will show that p(x) q(x) = q(x) p(x), ∀q(x)∈H×H [x; σ]. Without loss of its generality, the proof will be done for two cases q(x). They are q(x) = (a, b)∈H×H and q(x) = (1, 1)x.

Case q(x) = (a, b). Let:

Then:

So:

(1)

On the other hand:

(2)

From Eq. 1 and 2, we can see that p(x) q(x) = q(x) p(x). Case q(x) = (1, 1) x:

(3)

On the other hand:

(4)

From Eq. 3 and 4, we can see that p(x) q (x) = q(x) p(x.) From the two cases q(x) above it proves that p(x) q(x) = q(x) p(x), then such that p⊆Z (H×H[x; σ]). Second, we show that Z(H×H [x; σ])⊆P. Let, P(x)∈Z (H×H [x; σ]), so, p(x) holds p(x) q(x) = q(x) p(x), ∀q(x) ∈H×H [x; σ] and p(x) By choosing q(x) = (j, i) then:

(5)

Furthermore:

(6)

 

Because p(x) q(x) = q(x) p(x) then (Eq. 5) = (Eq. 6), we get:

Let:

And:

Then:

So, we have:

Let:

Then:

So, from this equation, we get:

And:

Let:

Then:

So, from this equation we get:

And:

Let:

Then:

So, from this equation we get:

And:

Let:

Then:

So, from this equation we get:

And:

Let:

Then:

So, from this equation, we get:

And:

Let:

Then:

So, from this equation we get:

And:

Based on the results above, it can be concluded that:

For coefficients x⁶ⁿ can be written as:

For coefficients x⁶ⁿ⁺¹ can be written as:

For coefficients x⁶ⁿ⁺² can be written as:

For coefficients x⁶ⁿ⁺³ can be written as:

For coefficients x⁶ⁿ⁺⁴ can be written as:

For coefficients x⁶ⁿ⁺⁵ can be written as:

Hence, p(x) can be written by:

Choose q(x) = (i, j), then:

So:

(7)

Furthermore:

(8)

Because p(x) q(x) = (x) p(x) then (Eq. 7) = pers (Eq. 8), we get:

Then:

Then:

Then:

 

Then:

Then:

Then:

 

So that:

Where:

Therefore, p(x)∈P, it show that Z(H×H [x; σ])⊆P because P⊆Z (H×H [x; σ]) and Z(H×H [x; σ])⊆P then it proves that Z(H×H [x; σ]) = P.

CONCLUSION

In this study, will describe the center forms of skew polynomial ring over a couple of quaternions.

How to cite this article:

Iuliia Iakovleva. The Sociological Typology of New Religious Communities.
DOI: https://doi.org/10.36478/rjagr.2020.7.10
URL: https://www.makhillpublications.co/view-article/1815-9354/rjagr.2020.7.10