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ISSN: 2350-0328 International Journal of Advanced Research in Science, Engineering and Technology Vol. 6, Issue 10 , October 2019 Review on Euclidean Domains, Principal Ideal Domains and Unique Factorisation Domains with its Applications Jaspreet kaur PG student, Assistant Professor, Department of Mathematics, Sri Guru Teg Bahadur Khalsa College, Sri Anandpur Sahib,Punjab, India. ABSTRACT: The aim of present paper is to review some results on ring theory. Throughout this paper we discuss Ɍ is commutative ring with unity. Particularly, we discuss the basic definitions of Rings, Ideals, Integral Domains, Principal Integral Domain (PID), Unique Factorization Domain(UFD) and Euclidean Domain(ED).We have presented some important application on Principal Ideal Domain (PID),Unique Factorization Domain(UFD) and Euclidean Domain(ED). KEYWORDS: Binary operations, Commutative Ring, Integral Domains, Ideals, Ring of Integers, Polynomial Rings etc. I.INTRODUCTION Covers some very basic concepts of Ring Theory over some fields like Integers, Rationals, Real numbers,Complex numbers etc. We begin with some basic definitions of Ring theory. As we study that Group theory stands only on the study of only one Binary operation, While Ring theory involves two Binary operations with additional elementry properties. There are many mathematical structures of study having two binary operations ,one is addition and the other is multiplication on ℤ, ℚ, Ɍ, ℂ, ℤ 푛ℤ, 푛 × 푛 푚푎푡푟푖푐푒푠 etc In Ring theory there is a very interested concept of “division”, ” division algorithm”, ” prime “, ”factorization” etc , from the ring of integers ℤ to an arbitrary commutative integral domain Ɍ with unity. Also we rephrase by definitions with innumerable examples ofIdeals, Divisors, Domains like Unique Factorization Domain,Principal Ideal Domain,Euclidean Domain along with their Applications including Euclidean Domain and the Gaussian Integers, Eisenstein integers, Polynomial rings, Smith Normal Form. II.LITERATURE SURVEY A) P.B.Bhattacharya, S.K. Jain, S.R.Nagpaul:- Outlined the expansion of ring theory with the concept of Ideals and homomorphism. In First edition he deals with rings and cover very basic concepts of rings, illustrated by numerous examples, including prime ideals, maximal ideals, UFD, PID and ED. Among these in second edition he represent the application division algorithm, Euclidean algorithm of Euclidean Domain(ED), Polynomial rings of Unique Factorization Domain(PID), Smith normal form of Principal ideal domain(PID). B) C Musili:- In Second Revised edition covering the very basic aspects of Rings, Ideals, Factorisation in Commutative Integral Domain. He also extended the concepts of Division, Division algorithm, Gaussian integers, Prime, Factorization ,Polynomial rings over PID etc from the ring of integers ℤ to an arbitrary commutative integral domain Ɍ with unity. C) David Joyce:-Expressed the ring theory its properties, Integral Domains, the Gaussian integers, Divisibility in Integral Domain, Euclidean Algorithm, Division for Polynomial that is Division Algorithm and also give very important Unique Factorization theorem proving result of Unique factorization domain. D) Linda Gilbert/Jimmi Gilbert:-She has been writing text book since 1981 with her husband Jimmi Gilbert including „Elements of Modern Algebra‟ and „Linear Algebra and Matrix theory‟. As the earlier editions the author gradually introduce and develop concepts to help make the material more accessible. In 7th edition developed the concept that how ring of integers working in Integral Domains, how a Field and Integral domain(finite /infinite) working together. Copyright to IJARSET www.ijarset.com 11044 ISSN: 2350-0328 International Journal of Advanced Research in Science, Engineering and Technology Vol. 6, Issue 10 , October 2019 E) Joseph J. Rotman:-The first edition is printed in 2002 and second in 2003.He study advanced algebra and its related topics .Introduces prime and maximal ideals in commutative rings, UFD etc. F) E Weiss, MC Grawhill:-One can initiate the study of algebraic number theory either globally and locally i.e. either by considering ideals in the rings of integers of number fields or else by looking first at the behaviour of field extensions at a single prime divisor and investing relationship among different prime of same field and gives results on Minkowski bound . III.RINGS A nonempty set Ɍ ≠ ∅ together with two binary operations „+‟,‟∙‟which are called as addition and multiplication (product) respectively, then it becomes a ring Ɍ if its satisfies following properties: (Ɍ, +) is an abelian group. (Ɍ,∙) is a semi group Distributive law hold for both left and right sides i.e. 푝(푞 + 푟) = 푝. 푞 + 푝. 푟 푎푛푑 푝 + 푞 푟 = 푝. 푟 + 푞. 푟 ∀푝, 푞, 푟휖Ɍ A. Ring With Unity: A ring(Ɍ, +,∙) in which multiplicative semi group has an identity element i.e.1 ∈ Ɍsuch that 푝. 1 = 푝 = 1. 푝 ∀푝 ∈ Ɍ. B. Commutative ring: A ring (Ɍ, +,∙) in which multiplicative semi group satisfies commutative property i.e. 푝. 푞 = 푞. 푝 ∀푝, 푞 ∈ Ɍ. C. Examples(Trivial and Non trivial) The rings ℤ, ℚ, Ɍ, ℂ are trivial examples of commutative ring with unity. An example of a non-trivial commutative ring in which every element is of square 0 (C Musili) different comparisons are: An example: 푀2 ℤ 2ℤ with usual addition but multiplication *defined as 푃 ∗ 푄 = 푃푄 + 푄푃 , but associativity of * is failure here. If we tried ℤ 2ℤ × ℤ 2ℤ × ℤ 2ℤ with point wise addition and multiplication in vector calculus. But this process is also not associative. IV. IDEAL Let Ɍ be a ring, Ҡ ⊂ Ɍ is a left ideal (right ideal) if it satisfies following two conditions: 푎, 푏 ∈ Ҡ → 푎 − 푏 ∈ Ҡ 푎 ∈ Ҡ 푎푛푑 푟 ∈ Ɍ → 푟푎 ∈ Ҡ 푎 ∈ Ҡ 푎푛푑 푟 ∈ Ɍ → 푎푟 ∈ Ҡ Here a Left Ideal or Right Ideal is a subring of(Ɍ, +, . ). An Ideal is also called a two sided ideal if the subset Ҡ of Ɍ is both of left and right ideal. A. Examples (Trivial and Non Trivial): 0 and Ɍ are the trivial examples of ideals of Ɍ 푝 푞 푝 0 The sets Ԍ = | 푝, 푞 ∈ Ɍ and Ԍ = | 푝, 푞 ∈ Ɍ are respectively right and left ideals of 1 0 0 2 푞 0 푀2 Ɍ for any ring Ɍ. As we know that proper means “푛표푡 Ɍ“ and nonzero means “푛표푡 0 and nontrivial means "푛표푡Ɍ 푎푛푑 푛표푡 {0}“ If 휉: Ɍ → Տ is ring homomorphism then its kernal푘푒푟휉 = 푟 ∈ Ɍ: 휉 푟 = 0 is an ideal of Ɍ and must be proper ideal of Ɍbecause if 푘푒푟휉 = 0then휉 = 푖푑푒푛푡푖푐푎푙푙푦 0. B. Prime Ideal:Let Ɍ be a commutative ring .An Ideal Ҡis said to be a Prime ideal of Ɍ if 1. Ҡ ≠ Ɍ 2. 푝, 푞 ∈ Ɍ, 푝푞 ∈ Ҡ → 푒푖푡ℎ푒푟 푝 ∈ Ҡ 표푟 푞 ∈ Ҡ C. Maximal left Ideals: Let Ɍ be a commutative ring then aleft ideal is called maximal ideal of Ɍ if 1. Ҡ ≠ Ɍ 2. For any ideal there is no Ideal strictly between Ҡ 푎푛푑 Ɍ 푖. 푒 If any ideal ℒ 표푓 Ɍ 푠푢푐ℎ 푡ℎ푎푡 Ҡ ⊆ ℒ ⊆ Ɍ ⇒ 푒푖푡ℎ푒푟 Ҡ = ℒ 표푟 ℒ = Ɍ is hold D.Minimal left Ideal:Let Ɍ be a commutative ring then aleft ideal is called minimal ideal of Ɍ if 1. Ҡ ≠ 0 2. For any ideal there is no ideal strictly between 0 푎푛푑 Ҡ i.e. Copyright to IJARSET www.ijarset.com 11045 ISSN: 2350-0328 International Journal of Advanced Research in Science, Engineering and Technology Vol. 6, Issue 10 , October 2019 0 ⊆ ℒ ⊆ Ҡ ⇒ 푒푖푡ℎ푒푟 ℒ = 0 표푟 Ҡ = 퓛 V. BASIC ALGEBRA OF THESE IDEALS A. Addition of Ideals:IfҠ 푎푛푑 ℒ are ideals in ring Ɍ ,then Ҡ + ℒ = x + y x ∈ Ҡ, y ∈ ℒ ⊆ Ɍ B. Multiplication of Ideals: IfҠ , ℒ are two ideals of the ringɌ, we define multiplication of two subsets Ҡ, ℒ 표푓 Ɍ as Ҡℒ = x1y1 + x2y2 + ⋯ xn yn /xiϵҠˎyi ∈ ℒˎ1 ≤ i ≤ nˎn ∈ N i.e. finite sums of 푥. VI. UNITS AND ZERO DIVISORS A. Units: Let Ɍ be a ring with unity .An element 푝 ∈ Ɍ is said to be a unit or invertible if ∃ 푞 ∈ Ɍ such that 푝푞 = 푞푝 = 1 the element 푞 is called the multiplicative inverse of 푝 and is denoted by 푝−1 . B. Examples: The rings ℤ, ℚ, Ɍ, ℂ are commutative ring with unity .Every non zero element of ℚ, Ɍ, ℂ is invertible and the inverse of 푝 is 푝−1 . However the only units in is ±1. C. Zero Divisors: An element 푝 ∈ Ɍ is said to be a left zero divisor if ∃푞 ≠ 0 suchthat 푝푞 = 0.Similarly 푝 is right zero divisor if ∃ 푟 ≠ 0 suchthat푟푝 = 0. An element 푝 ∈ Ɍ is zero divisors, if it is both left and right zero divisor. D. Example based on Units and Zero divisors: Find Zero divisors and Units of Ring ℤ푛 × ℤ푚 . The units of ℤ푛 and ℤ푚 are the different combinations of units of ℤ푛 and units of ℤ푚 If we take ℤ6 × ℤ2 then units are different combinations of units of ℤ6 (units are 1 and 5) and units of ℤ2 (unit is 1). In a finite commutative ring,a non zero element is either a unit or a zero divisors .If we find all elements of ℤ푛 × ℤ푚 and find all units , then the zero divisors are left i.e. not exist. VII. FIELDS A field is also a special type of ring .Let F be a ring with some additional properties is become to form a field. Conditions are: F is commutative ring. Identity element in F, i.e. e ≠ 0 Multiplicative inverse exist for every non zero element The Rationals, Reals, and Complex form field and also if any ring corresponding to any prime ℤ푝 is a field. VIII. INTEGRAL DOMAIN Firstly we revise that there are many ID like every field is an Integral Domain that is ℝi.e Real no‟s field ,Rings of Polynomials, The ring of Integers but the ring of integers which is IDgives very usable properties for Domains.
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  • Unit and Unitary Cayley Graphs for the Ring of Gaussian Integers Modulo N

    Unit and Unitary Cayley Graphs for the Ring of Gaussian Integers Modulo N

    Quasigroups and Related Systems 25 (2017), 189 − 200 Unit and unitary Cayley graphs for the ring of Gaussian integers modulo n Ali Bahrami and Reza Jahani-Nezhad Abstract. Let [i] be the ring of Gaussian integers modulo n and G( [i]) and G be the Zn Zn Zn[i] unit graph and the unitary Cayley graph of Zn[i], respectively. In this paper, we study G(Zn[i]) and G . Among many results, it is shown that for each positive integer n, the graphs G( [i]) Zn[i] Zn and G are Hamiltonian. We also nd a necessary and sucient condition for the unit and Zn[i] unitary Cayley graphs of Zn[i] to be Eulerian. 1. Introduction Finding the relationship between the algebraic structure of rings using properties of graphs associated to them has become an interesting topic in the last years. There are many papers on assigning a graph to a ring, see [1], [3], [4], [5], [7], [6], [8], [10], [11], [12], [17], [19], and [20]. Let R be a commutative ring with non-zero identity. We denote by U(R), J(R) and Z(R) the group of units of R, the Jacobson radical of R and the set of zero divisors of R, respectively. The unitary Cayley graph of R, denoted by GR, is the graph whose vertex set is R, and in which fa; bg is an edge if and only if a − b 2 U(R). The unit graph G(R) of R is the simple graph whose vertices are elements of R and two distinct vertices a and b are adjacent if and only if a + b in U(R) .