Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

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A Note on Hybrid Coset of a Nearring 

 

P. Narasimha Swamy 1a, Bhurgula Harika 1b, K. Vijay Kumar2c, B. Jyothi3d and K. 

Rajani1e 

1Department of Mathematics, School of Science, GITAM Deemed to be University, Hyderabad Campus-

502329, India 

2Department of Mathematics, KU College of Engineering and Technology, Kakatiya University, Warangal-

506009, India 

3Department of Mathematics, Telangana Mahila Vishwavidyalayam, Koti, Hyderabad-500095, India 

Email:    a swamy.pasham@gmail.com 

b harika.burgula84@gmail.com 

c vijay.kntm@gmail.com 

d jyothireddydumbala@gmail.com 

e rajanireddy2u@gmail.com 

 

Article History: 

Received: 30-01-2024 

Revised: 14-04-2024 

Accepted: 24-04-2024 

Abstract:  

The present study intends to learn hybrid coset of a nearring. It is explained with 

the adequate definitions and theorems of the hybrid coset of a nearing and near 

ring homomorphism. It has been demonstrated that hybrid coset of a nearring is 

a nearring epimorphism with kernel. Further, we established some important 

fundamental results in terms of hybrid structure corresponding to the nearrings. 

 

Keywords: Hybrid structure, Near ring, coset, Hybrid coset 

1. Introduction 

A nearring is an algebraic system connected to two binary operations, which gratifies all of a 

ring's axioms, with the conceivable exception of one distributive law. The very idea of nearring 

was first adapted by G. Pilz [1]. Nearrings have been the subject of investigation by 

Nobusawa[2], Bh Satyanarayana [3] and T. Srinivas [4]. In 1965, L.A. Zadeh [5] introduced 

new concept about fuzzy set. This concept highlighting the membership status of an 

indeterminate or fuzzy set. In this concept, membership status is defined as a function whose 

value is in the interval [0, 1]. Fuzzy set theory is established in many directions by many 

scholars and has got evoked great interest in the minds of many researchers who are working 

in different fields of mathematics. The theory of fuzzy sets has found many applications, 

including engineering, robotics design, computer modelling, and water resource planning. A 

hesitant fuzzy set is an extraordinary tool for disclosing people's hesitancy in every life and for 

handling with uncertainty, which could be suitably and matching way labelled in terms of the 

decision makers’ opinions. An extensive range of existing theories, like the probability theory, 

theory of fuzzy set, vague sets, interval mathematics theory, theory of rough set, etc., are 

observed to deal a variety of problems in many domains that which require data with 

uncertainties. V. Torra [6] established the perception of hesitant fuzzy sets. All these theories 



Communications on Applied Nonlinear Analysis 

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have their own limitations and difficulties which are elevated already [7]. To be free from these 

difficulties, D. Molodtsov [7] familiarized the soft set theory as a new mathematical tool for 

handling with uncertainties that is free from the difficulties. Molodtsov effectively applied the 

theory of soft set in many directions, such as functions’ smoothness, theory of game, Riemann 

integration, Perron integration operations research, probability, measurement theory and many 

other.  

As a parallel circuit of fuzzy sets and soft sets (or, hesitant fuzzy sets), Jun, Song and 

Muhiuddin [8] proposed the idea of hybrid structures in a set of parameters over an initial 

universe set, and illustrating numerous properties. Using this idea, they initiated the idea of a 

hybrid gamma near ring, hybrid ideal of a gamma near ring. B. Elavarasan [9] deliberated 

hybrid structures applied to ideals in near-rings. Saima Anis [10] has explored hybrid ideals in 

semigroups. M. Himaya Jaleela Begum [11] explored   hybrid fuzzy bi-ideals in near-rings. S. 

Abou-zaid [12] and S.D. Kim [13] were developed fuzzy ideals of near rings. P. Narasimha 

swamy [14] has developed sim of fuzzy ideals of Γ-near-rings. K. Vijay Kumar [15] has 

proposed the idea on bipolar fuzzy quasi ideals and bipolar N-subgroups of Near rings. 

Satyanarayana Bhavanari [16] has explored on fuzzy cosets of gamma near rings. T. Srinivas 

[17], Harika Bhurgula [18] and B. Jyothi [19,20] have established the concepts on near algebra. 

In which I have inspired to study on near ring concepts. In the current study, we acquaint with 

the conception of hybrid coset of a near ring and hybrid structure is used to analyze the 

structural statements of near rings. All over this paper 𝑁 means a (right) near ring. 

2. Preliminaries 

Definition 2.1: [8] Let 𝑈 be a universal set, 𝑃(𝑈) be the power set, 𝐿 be the set of parameters 

and 𝐼 be the unit interval. A mapping 𝜉𝜆 ≔ (𝜉 ̃, 𝜆): 𝐿 → 𝑃(𝑈) 𝑋 𝐼, 𝑞 → (𝜉(𝑞), 𝜆(𝑞)) i.e., the 

image of 𝑞 is chosen by (𝜉(𝑞), 𝜆(𝑞))  is entitled a hybrid structure (HS) in 𝐿 over 𝑈,  

where 𝜉: 𝐿 → 𝑃(𝑈) and 𝜆 ∶ 𝐿 → 𝐼 are the mappings. 

 

Definition 2.2: [8] Let 𝜉𝜆 be a HS in 𝐿 over 𝑈. Then the sets 

 𝜉𝜆[𝛼, 𝑡] = {𝑞 ∈ 𝐿 𝜉(𝑞)⁄ ⊇ 𝛼, 𝜆(𝑞) ≤ 𝑡}, 

 𝜉𝜆(𝛼, 𝑡] = {𝑞 ∈ 𝐿 𝜉(𝑞)⁄ ⊋ 𝛼, 𝜆(𝑞) ≤ 𝑡}, 

 𝜉𝜆[𝛼, 𝑡) = {𝑞 ∈ 𝐿 𝜉(𝑞)⁄ ⊇ 𝛼, 𝜆(𝑞) < 𝑡}, 

 𝜉𝜆(𝛼, 𝑡) = {𝑞 ∈ 𝐿 𝜉(𝑞)⁄ ⊋ 𝛼, 𝜆(𝑞) < 𝑡} 

are entitled the [𝛼, 𝑡] – hybrid cut (HC), (𝛼, 𝑡] – HC, [𝛼, 𝑡) – HC, and (𝛼, 𝑡) – HC of 𝜉𝜆 

respectively, where 𝛼 ∈ 𝑃(𝑈), 𝑡 ∈ 𝐼. Obviously, 𝜉𝜆(𝛼, 𝑡) ⊆  𝜉𝜆(𝛼, 𝑡] ⊆ 𝜉𝜆[𝛼, 𝑡] and 

𝜉𝜆(𝛼, 𝑡) ⊆ 𝜉𝜆[𝛼, 𝑡) ⊆ 𝜉𝜆[𝛼, 𝑡]. 

 



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Definition 2.3: [9] Let �̃�𝛾 be a HS of a nearring 𝑁, �̃�𝛾 is called a hybrid nearring of 𝑁 over 𝑈 

if the following conditions clutch: 

(𝑖)�̃�(𝑞 − 𝜍) ⊇ �̃�(𝑞) ∩ �̃�(𝜍), 𝛾(𝑞 − 𝜍) ≤ ⋁{𝛾(𝑞), 𝛾(𝜍)} ∀ 𝑞, 𝜍 ∈ 𝑁 

(𝑖𝑖)�̃�(𝑞𝜍) ⊇ �̃�(𝑞) ∩ �̃�(𝜍), 𝛾(𝑞𝜍) ≤ ⋁{𝛾(𝑞), 𝛾(𝜍)} ∀ 𝑞, 𝜍 ∈ 𝑁. 

3. Main Results 

In this, we introduce hybrid coset of a nearring (HCNR) and attain some of the properties 

of hybrid coset of a nearring. 

Definition 3.1: Let N be a NR. A HS �̃�𝛾 in 𝑁 over 𝑈 is entitled a HINR if the following 

conditions clutch. 

(𝑖) �̃�(𝑞 − 𝜍) ⊇ �̃�(𝑞) ∩ �̃�(𝜍), 𝛾(𝑞 − 𝜍) ≤ ⋁{𝛾(𝑞), 𝛾(𝜍)} ∀ 𝑞, 𝜍 ∈ 𝑁 

(𝑖𝑖) �̃�(𝑞𝜍) ⊇ �̃�(𝑞) ∩ �̃�(𝜍), 𝛾(𝑞𝜍) ≤ ⋁{𝛾(𝑞), 𝛾(𝜍)} ∀ 𝑞, 𝜍 ∈ 𝑁 

(𝑖𝑖𝑖)�̃�(𝑞 + 𝜍 − 𝑞) ⊇ �̃�(𝜍), 𝛾(𝑞 + 𝜍 − 𝑞) ≤ 𝛾(𝜍) ∀ 𝑞, 𝜍 ∈ 𝑁 

(𝑖𝑣)�̃�(𝑞𝜍) ⊇ �̃�(𝑞), 𝛾(𝑞𝜍) ≤ 𝛾(𝑞) ∀ 𝑞, 𝜍 ∈ 𝑁 

(𝑣)�̃�(𝜍(𝑞 + 𝑖) −  𝜍𝑞) ⊇ �̃�(𝑖), 𝛾(𝜍(𝑞 + 𝑖) −  𝜍𝑞) ≤ 𝛾(𝑖) ∀ 𝑞, 𝜍, 𝑖 ∈ 𝑁. 

If �̃�𝛾 gratifies (𝑖), (𝑖𝑖), (𝑖𝑖𝑖) and (𝑖𝑣) then �̃�𝛾 is entitled a right HINR of 𝑁. 

If �̃�𝛾 gratifies (𝑖), (𝑖𝑖), (𝑖𝑖𝑖) and (𝑣) then �̃�𝛾 is entitled a left HINR of 𝑁. 

Example 3.2: Let 𝑁 = {0, 𝑎𝜏, 𝑏𝜏, 𝑐𝜏} be a set with two binary operations ‘+’, ‘.’ as follows 

+ 0 𝑎𝜏 𝑏𝜏 𝑐𝜏 

0 0 𝑎𝜏 𝑏𝜏 𝑐𝜏 

𝑎𝜏 𝑎𝜏 0 𝑐𝜏 𝑏𝜏 

𝑏𝜏 𝑏𝜏 𝑐𝜏 0 𝑎𝜏 

𝑐𝜏 𝑐𝜏 𝑏𝜏 𝑎𝜏 0 

 

Then (𝑁, +, . ) is a nearring. Then the hybrid structure �̃�𝛾 in 𝑁 over 𝑈 = {𝑢1, 𝑢2, 𝑢3, 𝑢4, 𝑢5} 

which is given below 

 

 

 

 

 

 

Therefore (�̃�𝛾, 𝑁) is a HINR. 

. 0 𝑎𝜏 𝑏𝜏 𝑐𝜏 

0 0 0 0 0 

𝑎𝜏 𝑎𝜏 𝑎𝜏 𝑎𝜏 𝑎𝜏 

𝑏𝜏 𝑏𝜏 𝑏𝜏 𝑏𝜏 𝑏𝜏 

𝑐𝜏 𝑐𝜏 𝑐𝜏 𝑐𝜏 𝑐𝜏 

𝑁 �̃� 𝛾 

0 {𝑢1, 𝑢2, 𝑢3, 𝑢4, 𝑢5} 0.5 

𝑎𝜏 {𝑢1, 𝑢5} 0.6 

𝑏𝜏 {𝑢2, 𝑢3, 𝑢5} 0.7 

𝑐𝜏 {𝑢2, 𝑢4, 𝑢5} 0.8 



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Definition 3.3: Let �̃�𝛾 be a HINR of 𝑁 over 𝑈 and 𝜍 ∈ 𝑁. Then the hybrid coset (or coset) of 

�̃�𝛾 is denoted by 𝜍 + �̃�𝛾 and is demarcated by (𝜍 + �̃�)(𝑞) = �̃�(𝑞 − 𝜍) and (𝜍 + 𝛾)(𝑞) = 𝛾(𝑞 −

𝜍) ∀ 𝑞 ∈ 𝑁. 

Theorem 3.4: Let �̃�𝛾 be a HINR of 𝑁 over 𝑈 and 𝑞, 𝜍 ∈ 𝑁. Then 𝑞 + �̃�𝛾 = 𝜍 + �̃�𝛾 if and only 

if �̃�(𝑞 − 𝜍) = �̃�(0) and 𝛾(𝑞 − 𝜍) = 𝛾(0). 

Proof: Let 𝑞, 𝜍 ∈ 𝑁. Suppose that 𝑞 + �̃�𝛾 = 𝜍 + �̃�𝛾. Then �̃�(𝑞 − 𝜍) = (𝜍 + �̃�)(𝑞) =

(𝑞 + �̃�)(𝑞) = �̃�(𝑞 − 𝑞) = �̃�(0) and 𝛾(𝑞 − 𝜍) = (𝜍 + 𝛾)(𝑞) = (𝑞 + 𝛾)(𝑞) = 𝛾(𝑞 − 𝑞) =

𝛾(0). 

Conversely, suppose that �̃�(𝑞 − 𝜍) = �̃�(0) and 𝛾(𝑞 − 𝜍) = 𝛾(0). For every 𝜅 ∈ 𝑁, we have 

(𝑞 + �̃�)(𝜅) = �̃�(𝜅 − 𝑞) = �̃�(𝜅 − 𝜍 + 𝜍 − 𝑞) = �̃�[(𝜅 − 𝜍) + (𝜍 − 𝑞)] ⊇ �̃�(𝜅 − 𝜍) ∩ �̃�(𝜍 −

𝑞) = �̃�(𝜅 − 𝜍) ∩ �̃�(𝑞 − 𝜍) = �̃�(𝜅 − 𝜍) ∩ �̃�(0) = �̃�(𝜅 − 𝜍) =  (𝜍 + �̃�)(𝜅) and 

(𝑞 + 𝛾)(𝜅) = 𝛾(𝜅 − 𝑞) = 𝛾(𝜅 − 𝜍 + 𝜍 − 𝑞) = 𝛾[(𝜅 − 𝜍) + (𝜍 − 𝑞)]

≤∨ {𝛾(𝜅 − 𝜍), 𝛾(𝜍 − 𝑞)} =∨ {𝛾(𝜅 − 𝜍), 𝛾(𝑞 − 𝜍)} =∨ {𝛾(𝜅 − 𝜍), 𝛾(0)}

= 𝛾(𝜅 − 𝜍) = (𝜍 + 𝛾)(𝜅). 

Thus 𝑞 + �̃� ⊇ 𝜍 + �̃� and (𝑞 + 𝛾) ≤ (𝜍 + 𝛾). Now, (𝜍 + �̃�)(𝜅) = �̃�(𝜅 − 𝜍) = �̃�(𝜅 − 𝑞 + 𝑞 −

𝜍) = �̃�[(𝜅 − 𝑞) + (𝑞 − 𝜍)] ⊇ �̃�(𝜅 − 𝑞) ∩ �̃�(𝑞 − 𝜍) = �̃�(𝜅 − 𝑞) ∩ �̃�(𝑞 − 𝜍) = �̃�(𝜅 − 𝑞) ∩

�̃�(0) = �̃�(𝜅 − 𝑞) =  (𝑞 + �̃�)(𝜅) and (𝜍 + 𝛾)(𝜅) = 𝛾(𝜅 − 𝜍) = 𝛾(𝜅 − 𝑞 + 𝑞 − 𝜍) = 𝛾[(𝜅 −

𝑞) + (𝑞 − 𝜍)] ≤ ∨ {𝛾(𝜅 − 𝑞), 𝛾(𝑞 − 𝜍)} =∨ {𝛾(𝜅 − 𝑞), 𝛾(0)} = 𝛾(𝜅 − 𝑞) = (𝑞 + 𝛾)(𝜅). 

Thus 𝜍 + �̃� ⊇ 𝑞 + �̃� and (𝜍 + 𝛾) ≤ (𝑞 + 𝛾). Hence  𝑞 + �̃� = 𝜍 + �̃� and 𝑞 + 𝛾 = 𝜍 + 𝛾. 

Theorem 3.5: Let �̃�𝛾 be a HINR of 𝑁 over 𝑈. Then the following two statements hold: 

 If 𝑞 + �̃� = 𝑛 + �̃�,   𝜍 + �̃� = 𝑣 + �̃� then (𝑞 + 𝜍) + �̃� =  (𝑛 + 𝑣) + �̃� , 𝑞𝜍 + �̃� = 𝑛𝑣 + �̃� and 

 if 𝑞 + 𝛾 = 𝑛 + 𝛾,   𝜍 + 𝛾 = 𝑣 + 𝛾 then (𝑞 + 𝜍) + 𝛾 =  (𝑛 + 𝑣) + 𝛾 ,  

𝑞𝜍 + 𝛾 = 𝑛𝑣 + 𝛾 ∀ 𝑞, 𝜍, 𝑛, 𝑣 ∈ 𝑁. 

Proof: Suppose that 𝑞 + �̃� = 𝑛 + �̃�,   𝜍 + �̃� = 𝑣 + �̃� and 𝑞 + 𝛾 = 𝑛 + 𝛾,   𝜍 + 𝛾 = 𝑣 + 𝛾. 

Then �̃�(𝑞 − 𝑛) = �̃�(0), �̃�(𝜍 − 𝑣) = �̃�(0) and 𝛾(𝑞 − 𝑛) = 𝛾(0), 𝛾(𝜍 − 𝑣) = 𝛾(0). Consider    

�̃�[(𝑞 + 𝜍) − (𝑛 + 𝑣)]  = �̃�[(𝑞 − 𝑛) + (𝜍 − 𝑣)] ⊇ �̃�(𝑞 − 𝑛) ∩ �̃�(𝜍 − 𝑣) = �̃�(0) ∩ �̃�(0) =

�̃�(0) 

and 𝛾[(𝑞 + 𝜍) − (𝑛 + 𝑣)]  = 𝛾[(𝑞 − 𝑛) + (𝜍 − 𝑣)] ≤ ∨ {𝛾(𝑞 − 𝑛), 𝛾(𝜍 − 𝑣)} =∨

{𝛾(0), 𝛾(0)} = 𝛾(0). But �̃�(0) ⊇ �̃�[(𝑞 + 𝜍) − (𝑛 + 𝑣)] and 𝛾(0) ≤ 𝛾[(𝑞 + 𝜍) − (𝑛 + 𝑣)]. 

Therefore �̃�[(𝑞 + 𝜍) − (𝑛 + 𝑣)] = �̃�(0) and 𝛾[(𝑞 + 𝜍) − (𝑛 + 𝑣)]  = 𝛾(0).  

Thus (𝑞 + 𝜍) + �̃� = (𝑛 + 𝑣) + �̃� , 𝑞𝜍 + �̃� = 𝑛𝑣 + �̃� and (𝑞 + 𝜍) + 𝛾 =  (𝑛 + 𝑣) + 𝛾.  

Again �̃�[𝑞𝜍 − 𝑛𝑣]  = �̃�[𝑛𝑣 − 𝑞𝜍] = �̃�𝑟(𝑛𝑣 − 𝑞𝑣 + 𝑞𝑣 − 𝑞𝜍) 

                       = �̃�[(𝑛 − 𝑞)𝑣 + 𝑞(𝜍 + (−𝜍 + 𝑣)) − 𝑞𝜍] 

                               ⊇ �̃�((𝑛 − 𝑞)𝑣) ∩ �̃�(𝑞(𝜍 + (−𝜍 + 𝑣)) − 𝑞𝜍) 



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                                                           ⊇ �̃�(𝑛 − 𝑞) ∩ �̃�(−𝜍 + 𝑣) 

                                                           = �̃�(𝑛 − 𝑞) ∩ �̃�(𝜍 − 𝑣) 

                                                           = �̃�(0) ∩ �̃�(0) = �̃�(0) 

and      𝛾[𝑞𝜍 − 𝑛𝑣]  = 𝛾[𝑛𝑣 − 𝑞𝜍] = 𝛾(𝑛𝑣 − 𝑞𝑣 + 𝑞𝑣 − 𝑞𝜍) 

               = 𝛾[(𝑛 − 𝑞)𝑣 + 𝑞(𝜍 + (−𝜍 + 𝑣)) − 𝑞𝜍] 

                          ≤  ∨ {𝛾((𝑛 − 𝑞)𝑣), 𝛾(𝑞(𝜍 + (−𝜍 + 𝑣)) − 𝑞𝜍)} 

                                                       ≤ ∨ {𝛾(𝑛 − 𝑞), 𝛾(−𝜍 + 𝑣)} 

                                                       = ∨ {𝛾(𝑢 − 𝑞), 𝛾(𝜍 − 𝑣)} 

                                                       = ∨ {𝛾(0), 𝛾(0)} = 𝛾(0). 

But �̃�(0) ⊇ �̃�(𝑞𝜍 − 𝑛𝑣) and 𝛾(0) ≤ 𝛾(𝑞𝜍 − 𝑛𝑣). Therefore �̃�(𝑞𝜍 − 𝑛𝑣) = �̃�(0) and 𝛾(𝑞𝜍 −

𝑛𝑣)  = 𝛾(0). Thus 𝑞𝜍 + �̃� = 𝑛𝑣 + �̃� and 𝑞𝜍 + 𝛾 = 𝑛𝑣 + 𝛾. 

Notation 3.6: Let �̃�𝛾 be a HINR of 𝑁 over 𝑈. Then the set of all cosets of �̃�𝛾 is 

𝑁
�̃�𝛾

⁄ = {𝜍 + �̃�𝛾: 𝜍 ∈ 𝑁}, where 𝑁 �̃�⁄  = {𝜍 + �̃�: 𝜍 ∈ 𝑁} and 𝑁 𝛾⁄  = {𝜍 + 𝛾: 𝜍 ∈ 𝑁}. 

Theorem 3.7: Let �̃�𝛾 be a HINR of 𝑁 over 𝑈. Then 𝑁 �̃�𝛾
⁄  is a near ring with respect to the 

operations defined by  

(𝑞 + �̃�) + (𝜍 + �̃�) = (𝑞 + 𝜍) + �̃�, (𝑞 + �̃�)(𝜍 + �̃�) = (𝑞𝜍) + �̃�   and 

(𝑞 + 𝛾) + (𝜍 + 𝛾) = (𝑞 + 𝜍) + 𝛾, (𝑞 + 𝛾)(𝜍 + 𝛾) = (𝑞𝜍) + 𝛾 ∀ 𝑞, 𝜍 ∈ 𝑁. 

Proof: A direct verification shows that (𝑁
�̃�𝛾

⁄ , +) is a group. Let 𝑞 + �̃�, 𝜍 + �̃�, 𝑗 + �̃� ∈ 𝑁
�̃�𝛾

⁄  

and +𝛾, 𝜍 + 𝛾, 𝑗 + 𝛾 ∈ 𝑁
�̃�𝛾

⁄ , where 𝑞, 𝜍, 𝑗 ∈ 𝑁. Then [(𝑞 + �̃�)(𝜍 +  �̃�)](𝑗 +  �̃�) =

(𝑞𝜍 + �̃�)(𝑗 +  �̃�) = (𝑞𝜍)𝑗 + �̃� = 𝑞(𝜍𝑗) + �̃� = (𝑞 + �̃�)[(𝜍 +  �̃�)(𝑗 +  �̃�)] and [(𝑞 + 𝛾)(𝜍 +

 𝛾)](𝑗 +  𝛾) = (𝑞𝜍 + 𝛾)(𝑗 +  𝛾) = (𝑞𝜍)𝑗 + 𝛾 = 𝑞(𝜍𝑗) + 𝛾 = (𝑞 + 𝛾)[(𝜍 +  𝛾)(𝑗 +  𝛾)]. This 

shows that 𝑁 �̃�𝛾
⁄  is a semi group under multiplication. Consider 

 [(𝑞 + �̃�) + (𝜍 + �̃�)](𝑗 +  �̃�) = ((𝑞 + 𝜍) + �̃�)(𝑗 +  �̃�) = (𝑞 + 𝜍)𝑗 + �̃� 

                                                      = (𝑞𝑗 + 𝜍𝑗) + �̃� = (𝑞𝑗 + �̃�) + (𝜍𝑗 + �̃�) 

                                               = (𝑞 + �̃�)(𝑗 + �̃�) + (𝜍 + �̃�)(𝑗 +  �̃�), 

 [(𝑞 + 𝛾) + (𝜍 +  𝛾)](𝑗 +  𝛾) = ((𝑞 + 𝜍) + 𝛾)(𝑗 +  𝛾) = (𝑞 + 𝜍)𝑗 + 𝛾 

                                                      = (𝑞𝑗 + 𝜍𝑗) + 𝛾 = (𝑞𝑗 + 𝛾) + (𝜍𝑗 + 𝛾) 

                                                      = (𝑞 + 𝛾)(𝑗 +  𝛾) + (𝜍 +  𝛾)(𝑗 +  𝛾). 

Hence  𝑁 �̃�𝛾
⁄  is a nearring.    



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Definition 3.8: Let �̃�𝛾 be a HINR of 𝑁. Then 𝑁 �̃�𝛾
⁄ , the set of all cosets of �̃�𝛾 is called a 

hybrid quotient nearring of 𝑁 by �̃�𝛾 with respect to the following operations:  

(𝑞 + �̃�) + (𝜍 + �̃�) = (𝑞 + 𝜍) + �̃� and (𝑞 + 𝛾) + (𝜍 +  𝛾) = (𝑞 + 𝜍) + 𝛾, 

(𝑞 + �̃�)(𝜍 +  �̃�) = (𝑞𝜍) + �̃� and (𝑞 + 𝛾)(𝜍 +  𝛾) = (𝑞𝜍) + 𝛾 

for every 𝑞, 𝜍 ∈ 𝑁. 

Theorem 3.9: Let �̃�𝛾 be a HINR of 𝑁. Define ∅: 𝑁
�̃�𝛾

⁄ → 𝑃(𝑈)𝑋𝐼 by ∅(𝑞 + �̃�𝛾) = �̃�𝛾(𝑞)  

 i.e., ∅(𝑞 + �̃�) = �̃�(𝑞) and ∅(𝑞 + 𝛾) = 𝛾(𝑞) ∀ 𝑞 ∈ 𝑁. Then ∅ is a hybrid ideal of 𝑁 �̃�𝛾
⁄ . 

Proof: Suppose that 𝑞 + �̃� = 𝜍 +  �̃� and 𝑞 + 𝛾 = 𝜍 +  𝛾. Then �̃�(𝑞 − 𝜍) = �̃�(0) and 

𝛾(𝑞 − 𝜍) = 𝛾(0). This implies that �̃�(𝑞) = �̃�(𝜍) and 𝛾(𝑞) = 𝛾(𝜍) i.e., ∅(𝑞 + �̃�) = ∅(𝜍 + �̃�) 

and ∅(𝑞 + 𝛾) = ∅(𝜍 + 𝛾). Therefore ∅ is well defined. 

We verify that ∅ is a hybrid ideal of 𝑁 �̃�𝛾
⁄ . 

Let 𝑞 + �̃�𝛾, 𝜍 + �̃�𝛾, 𝑗 + �̃�𝛾 ∈ 𝑁
�̃�𝛾

⁄ . 

(𝑖) ∅((𝑞 + �̃�) + ( 𝜍 +  �̃�)) = ∅((𝑞 + 𝜍) + �̃�) = �̃�(𝑞 + 𝜍) ⊇ �̃�(𝑞) ∩ �̃�(𝜍) = ∅(𝑞 + �̃�) ∩

 ∅(𝜍 + �̃�), and ∅((𝑞 + 𝛾) + ( 𝜍 +  𝛾)) = ∅((𝑞 + 𝜍) + 𝛾) = 𝛾(𝑞 + 𝜍) ≤ ⋁{𝛾(𝑞), 𝛾(𝜍)} =

⋁{∅(𝑞 + 𝛾), ∅(𝜍 + 𝛾)}. 

(𝑖𝑖) ∅(𝑞 + �̃�) = �̃�(𝑞) = �̃�(−𝑞) = ∅(−𝑞 + �̃�) and ∅(𝑞 + 𝛾) = 𝛾(𝑞) = 𝛾(−𝑞) = ∅(−𝑞 + 𝛾). 

(𝑖𝑖𝑖) ∅(( 𝜍 + �̃�) + (𝑞 + �̃�) − ( 𝜍 + �̃�)) = ∅((𝜍 + 𝑞 − 𝜍) + �̃�) = �̃�(𝜍 + 𝑞 − 𝜍) = �̃�(𝑞) =

∅(𝑞 + �̃�) and ∅(( 𝜍 +  𝛾) + (𝑞 + 𝛾) − ( 𝜍 +  𝛾)) = ∅((𝜍 + 𝑞 − 𝜍) + 𝛾) = 𝛾(𝜍 + 𝑞 − 𝜍) =

𝛾(𝑞) = ∅(𝑞 + 𝛾). 

(𝑖𝑣) ∅(( 𝜍 +  �̃�)((𝑞 + �̃�) + (𝑗 + �̃�) − ( 𝜍 +  �̃�)(𝑞 + �̃�))

= ∅ (( 𝜍 + �̃�)((𝑞 + 𝑗) + �̃�) − (𝜍𝑞 + �̃�)) 

                           = ∅((𝜍(𝑞 + 𝑗) + �̃�) − (𝜍𝑞 + �̃�)) 

                           = ∅((𝜍(𝑞 + 𝑗) − 𝜍𝑞) + �̃�) = �̃�(𝜍(𝑞 + 𝑗) − 𝜍𝑞) = �̃�(𝑗) = ∅(𝑗 + �̃�) and  

∅(( 𝜍 +  𝛾)((𝑞 +  𝛾) + (𝑗 +  𝛾) − ( 𝜍 +   𝛾)(𝑞 +  𝛾))

= ∅ (( 𝜍 +   𝛾)((𝑞 + 𝑗) +  𝛾) − (𝜍𝑞 +  𝛾)) 

                           = ∅((𝜍(𝑞 + 𝑗) +  𝛾) − (𝜍𝑞 +  𝛾)) 

                           = ∅((𝜍(𝑞 + 𝑗) − 𝜍𝑞) +  𝛾) =  𝛾(𝜍(𝑞 + 𝑗) − 𝜍𝑞) =  𝛾(𝑗) = ∅(𝑗 +  𝛾). 

Hence ∅ is a hybrid ideal of 𝑁 �̃�𝛾
⁄ . 



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Definition 3.10: Let 𝑁1, 𝑁2 be near rings. A mapping ∅: 𝑁1 → 𝑁2 is called a nearing 

homomorphism if ∅(𝑞 + 𝜍) = ∅(𝑞) + ∅(𝜍) and ∅(𝑞𝜍) = ∅(𝑞)∅(𝜍) ∀ 𝑞, 𝜍 ∈ 𝑁1. Moreover if 

∅ is one-one then ∅ is called as monomorphism, if ∅ is onto then ∅ is called epimorphism, 

if ∅ is bijective then ∅ is called an isomorphism.  

Theorem 3.11: If �̃�𝛾 is a HINR 𝑁 then the mapping ∅: 𝑁 → 𝑁/�̃�𝛾 defined by ∅(𝑞) = 𝑞 +

�̃�𝛾 ∀ 𝑞 ∈ 𝑁 where ∅(𝑞) = 𝑞 + �̃� and  ∅(𝑞) = 𝑞 + 𝛾, is a near ring epimorphism with 

kernal �̃�𝛾∗
 where  

 �̃�𝛾∗
= {𝑞 ∈ 𝑁: �̃�𝛾(𝑞) = �̃�𝛾(0)} ie., �̃�𝛾∗

= {𝑞 ∈ 𝑁: �̃�(𝑞) = �̃�(0) 𝑎𝑛𝑑  𝛾(𝑞) =  𝛾(0)}. 

Proof: Let 𝑞, 𝑦 ∈ 𝑁. Suppose that 𝑞 = 𝜍. Then 𝑞 + �̃� = 𝜍 + �̃� and 𝑞 + 𝛾 = 𝜍 + 𝛾. This 

implies that ∅(𝑞) = ∅(𝜍). Therefore ∅ is well defined. 

Now ∅(𝑞 + 𝜍) = (𝑞 + 𝜍) + �̃� = (𝑞 + �̃�) + (𝜍 + �̃�) = ∅(𝑞) + ∅(𝜍) and  

∅(𝑞 + 𝜍) = (𝑞 + 𝜍) + 𝛾 = (𝑞 + 𝛾) + (𝜍 + 𝛾) = ∅(𝑞) + ∅(𝜍), 

∅(𝑞𝜍) = (𝑞𝜍) + �̃� = (𝑞 + �̃�)(𝜍 + �̃�) = ∅(𝑞)∅(𝜍) and ∅(𝑞𝜍) = (𝑞𝜍) + 𝛾 = (𝑞 + 𝛾)(𝜍 +

𝛾) = ∅(𝑞)∅(𝜍). 

Therefore ∅ is a homomorphism. 

Let 𝑞 + �̃�𝛾 ∈ 𝑁
�̃�𝛾

⁄ . Then 𝑞 ∈ 𝑁. For this 𝑞 ∈ 𝑁, we have ∅(𝑞) = 𝑞 + �̃�𝛾. Therefore ∅ is a 

near ring epimorphism. And now 𝑞 ∈ ker∅ ⇔ ∅(𝑞) = 0 = 0 + �̃� = 0 + 𝛾 

                                                                         ⇔  𝑞 + �̃� = 0 + �̃� and 𝑞 + 𝛾 = 0 +  𝛾 

⇔ �̃�(𝑞 − 0) = �̃�(0) and 𝛾(𝑞 − 0) = 𝛾(0) ⇔  �̃�(𝑞) = �̃�(0) and 𝛾(𝑞) = 𝛾(0) ⇔ 𝑞 ∈ �̃�𝛾∗
. 

This shows that kernal ∅ = �̃�𝛾∗
. 

 

Theorem 3.12: If �̃�𝛾 is a HINR 𝑁. Then 𝑁 �̃�𝛾
⁄  is isomorphic to 𝑁 �̃�𝛾∗

⁄ , 

where �̃�𝛾∗
= {𝑞 ∈ 𝑁: �̃�(𝑞) = �̃�(0) 𝑎𝑛𝑑  𝛾(𝑞) =  𝛾(0)}. 

Proof: We have that �̃�𝛾∗
 is an ideal(kernal) of 𝑁. We know that 𝑁 �̃�𝛾

⁄ = {𝑞 + �̃�𝛾: 𝑞 ∈ 𝑁} and  

𝑁
�̃�𝛾∗

⁄ = {𝑞 + �̃�𝛾∗
: 𝑞 ∈ 𝑁}. Define a mapping ∅: 𝑁

�̃�𝛾
⁄ → 𝑁

�̃�𝛾∗
⁄  by ∅(𝑞 + �̃�) = 𝑞 + �̃�∗ and 

∅(𝑞 + 𝛾) = 𝑞 + 𝛾∗ for every 𝑞 + �̃�, 𝑞 + 𝛾 ∈ 𝑁
�̃�𝛾

⁄ . To prove that 𝑁
�̃�𝛾

⁄  is isomorphic to 

𝑁
�̃�𝛾∗

⁄ , it is sufficient to prove that ∅ is well defined, one-one, onto and homomorphism. 

Let 𝑞 + �̃�𝛾, 𝜍 + �̃�𝛾 ∈ 𝑁
�̃�𝛾

⁄ , where 𝑞, 𝛾 ∈ 𝑁. 



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Then 𝑞 + �̃� = 𝜍 + �̃� ⇔ �̃�(𝑞 − 𝜍) = �̃�(0) ⇔ 𝑞 − 𝜍 ∈ �̃�∗ ⇔  𝑞 + �̃�∗ = 𝜍 + �̃�∗ ⇔ ∅(𝑞 + �̃�) =

∅(𝜍 + �̃�) and 𝑞 + 𝛾 = 𝜍 + 𝛾 ⇔ 𝛾(𝑞 − 𝜍) = 𝛾(0) ⇔ 𝑞 − 𝜍 ∈ 𝛾∗ ⇔  𝑞 + 𝛾∗ = 𝜍 + 𝛾∗ ⇔ 

∅(𝑞 + 𝛾) = ∅(𝜍 + 𝛾). Therefore ∅ is well defined and one-one. 

Let 𝜍 + �̃�𝛾∗
∈ 𝑁

�̃�𝛾∗
⁄ . Then 𝜍 ∈ 𝑁. For this 𝜍 ∈ 𝑁, we have 𝑞 + �̃�𝛾 ∈ 𝑁

�̃�𝛾
⁄  and ∅(𝜍 + �̃�) =

 𝜍 + �̃�∗ and ∅(𝜍 + 𝛾) =  𝜍 + 𝛾∗. That is for each 𝜍 + �̃�𝛾∗
∈ 𝑁

�̃�𝛾∗
⁄  there exists 𝜍 + �̃�𝛾 ∈ 𝑁

�̃�𝛾
⁄  

such that ∅(𝜍 + �̃�) =  𝜍 + �̃�∗ and ∅(𝜍 + 𝛾) =  𝜍 + 𝛾∗. Therefore ∅ is onto. 

Let 𝑞 + �̃�𝛾, 𝜍 + �̃�𝛾 ∈ 𝑁
�̃�𝛾

⁄ , 𝑞, 𝛾 ∈ 𝑁. Then ∅((𝑞 + �̃�) + (𝜍 + �̃�)) = ∅((𝑞 + 𝜍) + �̃�) = (𝑞 +

𝜍) + �̃�∗ = (𝑞 + �̃�∗) + (𝜍 + �̃�∗) = ∅(𝑞 + �̃�) + ∅(𝜍 + �̃�) and ∅((𝑞 + 𝛾) + (𝜍 + 𝛾)) = ∅((𝑞 +

𝜍) + 𝛾) = (𝑞 + 𝜍) + 𝛾∗ = (𝑞 + 𝛾∗) + (𝜍 + 𝛾∗) = ∅(𝑞 + 𝛾) + ∅(𝜍 + 𝛾), 

∅((𝑞 + �̃�)(𝜍 + �̃�)) = ∅((𝑞𝜍) + �̃�) = (𝑞𝜍) + �̃�∗ = (𝑞 + �̃�∗)(𝜍 + �̃�∗) = ∅(𝑞 + �̃�)∅(𝜍 + �̃�) 

and          ∅((𝑞 + 𝛾)(𝜍 + 𝛾)) = ∅((𝑞𝜍) + 𝛾) = (𝑞𝜍) + 𝛾∗ = (𝑞 + 𝛾∗)(𝜍 + 𝛾∗) = ∅(𝑞 +

𝛾)∅(𝜍 + 𝛾. ) 

Therefore ∅ is a homomorphism.  Hence 𝑁 �̃�𝛾
⁄  is isomorphic to 𝑁 �̃�𝛾∗

⁄ . 

4. Conclusion 

In this study, we familiarized the idea of hybrid coset of a nearring and explored numerous 

properties. Using these ideas, we familiarized the ideas of nearring homomorphism and 

isomorphism. Research can be prolonged to the hybrid ideal of a gamma near algebra, sum of 

hybrid ideals of a near algebra and sum of hybrid ideals of a gamma near algebra. 

5. Acknowledgements 

The authors outspread their appreciation to GITAM Deemed to be University, India for 

funding this research under GITAM SEED grant file number 2021/0104. 

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