Quotient Seminear-Rings

Home , Congruence relation, Ring homomorphism

ISSN (Print) : 0974-6846 Indian Journal of Science and Technology, Vol 9(38), DOI: 10.17485/ijst/2016/v9i38/89115, October 2016 ISSN (Online) : 0974-5645 Quotient Seminear-Rings

Fawad Hussain1, Muhammad Tahir2, Saleem Abdullah2 and Nazia Sadiq2 1Department of Mathematics, Abbottabad University of Science and Technology, Pakistan; [email protected] 2Department of Mathematics, Hazara University Mansehra, Pakistan; [email protected]; [email protected]; [email protected]

Abstract The study of a seminear-ring was started in 1967. Seminear-ring is the generalization of semiring and nearing. It is known that in a semiring the quotient structure is constructed by congruence relations through c-ideals and homomorphisms. We apply the said congruence relations to a seminear-ring and get quotient structure of a seminear-ring. The aim of this paper

quotientis to discuss seminear-rings. quotient seminear-rings in two different ways. We study congruence relations, homomorphisms and ideals. We show that each homomorphism and c-ideal define a congruence relation on seminear-rings. At the end, we discuss Keywords:

Seminear-ring, c-Ideals, Homomorphism, Congruence Relation, Quotient Seminear-Ring. 1. Introduction rings and explored some interesting properties. There is another intersecting paper7 in which the author discussed The study of semirings was started by the German seminear-rings as well as semi-near fields. Further differ- Mathematician Dedekind1. They were later studied by ent people in8-11 worked on seminear-rings and explored different Mathematicians, particularly by the American many interesting and elegant properties. The author of8,9 Mathematician Vandiver. He did work very hard on discussed substructures in seminear-rings. The author of11 semirings. He wanted to accept a semiring as fundamen- discussed radicals in seminear-rings while the authors’ tal and best algebraic structure2. He was not successful of10 discussed weekly regular seminear-rings. Before because of few reasons and semirings had fallen into four years ago Perumal and Balakrishnan discussed disuse. However, during the late 1960’s real and sig- left bipotent seminear-rings in12 and discussed left duo nificant applications were found of semirings in several seminear-rings in13 and explored some useful properties. fields. These fields include: automata theory, optimiza- In this paper, we study quotient structure of a seminear- tion theory, graph theory, the theory of discrete event ring while different quotient algebraic structures have dynamical system, coding theory, analysis of computer been studied in14,15. programs, algebras of formal processes and generalized fuzzy computation. The detail of these can be found 2. Preliminaries in3. Later on, different people worked on semirings and explored many interesting properties of semirings. In4 the This portion contains some of the basic definitions, fun- rough set theory has been applied to semirings while in5 damental results, and reviews some of the background some applications of a semiring have been studied. In the material, which will be used in the coming sections. In paper6 the authors introduced the notion of a seminear- 1967, Hoorn and Rootoselaar introduced the notion of a ring which was a generalization of a semiring and then semiring which was a generalization of a semiring. We explored some properties of seminear-rings. Especially start with the following definition of a semiring which has the authors of6 discussed homomorphisms in seminear- been taken from7.

*Author for correspondence Quotient Seminear-Rings

2.1 Definition (i) x + y S for all x, y S, (ii) x ⋅ y S for all x, y S. A seminear-ring is a non-empty set R together with two ∈ ∈ binary operations ‘+’ and ‘ ’ such that (R, +) and (R, ) are Proof ∈ ∈ semigroups such that The proof is easy and is left for the readers. (a + b) ⋅ c = (a ⋅ c) + (b ⋅ c) holds for all a, b, c R. We are now going to define ideals. The following definition has been taken from the paper8. It will be more suitable to call what we have just∈ defined a right seminear-ring. In the same way, we may define a 2.5 Definition left seminear-ring in which one use left distributive law instead of right distributive law. In this paper, we shall A non-empty subset I of a seminear-ring(R, +, ) is called keep to right seminear-rings throughout. To understand a left (respectively right) ideal ofR if the following condi- the notion of semirings, we give some examples. tions hold: (i) x + y ∈ I for all x, y ∈ I, 2.2 Examples (ii) a ⋅ x ∈ I (x ⋅ a ∈ I ) for all x ∈ I and a ∈ R. I is called a two-sided ideal or simply an ideal of R if it (i) All rings and semirings are seminear-rings. is both left as well as right ideal ofR . (ii) Let N be the set of natural numbers, then N is a Note that it is obvious from the above definition that semiring with ordinary addition and multiplication every ideal is a subseminear-ring and we know that every of numbers. subseminear-ring is a seminear-ring. Thus, it follows that We are now giving a non-trivial example of a semi- every ideal is a seminear-ring. near-ring. We are now going to define c-ideals in seminear- (iii) Let N be the set of natural numbers. Let rings. The idea of this definition comes from the paper16 a b  in which the authors define c-ideals in semirings. Ω =  : a, b, c, d ∈ N  c d    2.6 Definition and define, A+ B = AB and A ⋅ B = A for all A, B Ω . An ideal I of a seminear-ring(R, +, ⋅) is called a c-ideal Then ( Ω , +, ⋅) is a seminear-ring. of R if for any a, b ∈ R, there exist x, y ∈ I such that We are now going to define subseminear-rings and a + b + x = y + b + a. ideals in seminear-rings. Firstly we are going to define Like homomorphisms of other algebraic structures subseminear-rings for which we don’t have a specific such as semirings and rings, homomorphisms of semi- reference. near-rings are those maps which preserve the binary operations. We now give a proper definition of a semi- 2.3 Definition near-ring homomorphism. The following definition is taken from8. A non-empty subset S of a seminear-ring (R, +, ⋅) is called a subseminear-ring of R if S itself is a seminear-ring under the operations of R when restricted to S. 2.7 Definition It should be noted that very seminear-ring (R, +,⋅) has Let (R, +, ∙) and (S, , ) be two seminear-rings. A at least one subseminear-ring, i.e. R itself called the trivial mapping µ : R S from R to S is called a homomor- subseminear-ring. All other subseminear-rings are called phism if and only if: non-trivial subseminear-rings. We are now going to pres- (i) µ (a + b) = µ (a) µ(b) ent some properties of subseminear-rings. The result (ii) µ (a ⋅ b) = µ (a) µ(b) given below gives us necessary and sufficient conditions for all a, b R. for subseminear-rings and for which we don’t have a spe- It should be noted that if R = S, then the cific reference. homomorphism∈ µ is known as an endomorphism. If µ is homomorphism as well as onto, then µ is called an 2.4 Theorem epimorphism. If µ is homomorphism as well as one-one, Let (R, +, ⋅) be a seminear-ring, then a non-empty subset then µ is called a monomorphism. A homomorphism µ S of R is called a subseminear-ring of R if and only if is called an isomorphism if it is epimorphism as well as

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ρ monomorphism. An isomorphism µ is called an auto- Similarly one can show that is a right congruence morphism if R = S. If µ : R S is an isomorphism then relation. ρ we say that R is isomorphic to S and write: Conversely, suppose that is both left as well as right congruence relation. Choose s, t, u, v ∈R such that (s, t), R S. (u, v)∈ρ. This implies that s( + u, t + u), (s ∙ u, t ∙ u)∈ρ, because ρ is right compatible and (t + u, t + v), (t ∙ u, t ∙v)∈ρ, 3. Congruence Relations because ρ is left compatible. It follows that s( + u, t + v), (s ∙ u, t ∙ v)∈ρ, because ρ is transitive. This completes the This section concerned with the congruence relations on proof. seminear-rings. We define congruence relations and show that each homomorphism defines a congruence relation 4. Congruence Relations and on seminear-rings. The idea of this section comes from the book17 in which the author does similar calculations Homomorphisms for semigroups. In this section we state and prove a result which says that corresponding to every homomorphism there is a con- 3.1 Definition gruence relation. The result is important because once we Let (R, +, ∙) be a seminear-ring. A relation ρ on R is called get it we can get quotient seminear-rings. At the end we left compatible if for all s, t, a∈R such that (s, t) ∈ρ prove analogues of the isomorphism theorems. implies that (a + s, a + t), (a ∙ s, a ∙ t)∈ρ. ρ is called right compatible if for all s, t, b ∈R such 4.1 Theorem that (s, t) ∈ρ implies that (s + b, t + b), (s ∙ b, t ∙ b) ∈ ρ. If h is a homomorphism from a seminear-ring (R, +, ∙) ρ is called compatible if for all s, t, u, v∈R such that (s, to a seminear-ring (S, , ), then h defines a congru- t), (u, v) ∈ρimplies that (s + u, t + v), (s ∙u, t∙v)∈ρ. ence relation ρ on R given by (r, s) ρ if and only if A left (respectively right) compatible equivalence h (r) = h (s). relation is said to be a left (respectively right) congruence relation. A compatible equivalence relation is said to a Proof congruence relation. First we show that ρ is an equivalence relation. As h(r) = In order to understand the above concept, we give an h(r) for all r∈R, therefore (r, r) ρ and so the relation ρ example. is reflexive. If(r, s) ρ for some r, s R, then h (r) = h(s) and this implies that h(s) = h(r). Thus,(s, r) ρ and so 3.2 Example the relation ρ is symmetric. Now if (r, s), (s, t) ρ, then Consider the seminear-ring ( Ω , +, ⋅) of Example 2.2 (iii). according to the definitionh (r) = h (s) and h(s) = h (t) and Let = {(A, B): A = B} be a relation on Ω . Then one can this gives us h(r) =h(t). This implies that (r, t) ρ and so show that is a congruence relation on Ω . the relation ρ is transitive. Now let (r, s), (t, u) ρ, then 휌 We are now going to state and prove a result which according to the definition h(r) = h(s) and h(t) = h(u). 휌 gives equivalent conditions for congruence relations. Now h (r + t) = h(r) h (t) = h(s) h (u) = h(s + u). It follows that (r + t, s + u) ρ. In the same way one can 3.3 Proposition show that (r ⋅ t, s ⋅ u) ρ. Thus, the relation ρ is compatible. This completes the proof. A relation ρ on a seminear-ring R is a congruence relation Now let ρ be an equivalence relation on a set X. Then if and only if it is both left as well as aright congruence the equivalence class corresponding to the element x of relation. X is represented by the symbol x ρ and is defined as fol- Proof lows: Assume that ρ is a congruence relation on R. Choose s, t, xρ = {y ∈X (x,y)∈ρ}. a ∈R such that (s, t)∈ρ, then (a + s, a + t),(a ∙s, a ∙ t)∈ ρ, because (a, a)∈ρ. It follows that ρ is a left congruence If ρ is a congruence relation on a seminear-ring R, then relation. we say that xρ is called a congruence class corresponding

Vol 9 (38) | October 2016 | www.indjst.org Indian Journal of Science and Technology 3 Quotient Seminear-Rings

to the element x of R. Let R/ρ be the set of all congruence = yρ ⋅ xρ+zρ ⋅ xρ classes. That is Thus (R/ρ, +, ⋅) is a seminear-ring which is called a quotient seminear-ring. R/ρ = {xρ: x∈R}. We are now going to state and prove the seminear- We are now going to state a result which is taken from the ring analogues of the first, second and third isomorphism book18 and will be used later. The result is true for classes theorems. The semigroup equivalents may be found in but as we know that every class is a set, so in particular, it the book17. is true for sets as well. 4.3 Theorem [First Isomorphism Theorem] 4.2 Lemma Assume that ρ is a congruence relation on a seminear- Let X be a set and ρ an equivalence relation on X. Thenx ρ ring R. Then R/ρ is a seminear-ring under the following = y ρ if and only if (x, y)∈ρ. binary operations: Let x, y ∈R and xρ, yρ be the congruence classes xρ + yρ = (x + y)ρ and xρ ⋅ yρ = (x ⋅ y)ρ corresponding to the elements x and y, then we can define # for all xρ, yρ R/ρ. The mapping ρ : R →R/ρ binary operations on the quotient set R / ρin the following # defined by ρ (x) = xρ for all x∈R is an epimor- way: phism. Let (R, +, ∙) and (S, , ) be seminear-rings xρ+yρ= (x +y)ρ and xρ ⋅ yρ = (x ⋅ y)ρ. and Φ : R →S a homomorphism. Then the relation ker Φ = {(x, y) ∈R×R:Φ (x) = Φ (y)}is a congruence rela- These operations are well defined, since for allw , x, y, z ∈ tion on the seminear-ring R and there is a monomorphism ρ ρ ρ ρ R such that w = y and x = z , then by Lemma 4.2, (w, α : R/ker Φ →S such that ranα = ran Φ and the diagram ρ ρ ⋅ y)∈ and (x, z)∈ . This implies that (w + x, y + z), (w given below commutes. x, y ⋅ z)∈ρ, since ρ is a congruence relation. Thus, again Φ Lemma 4.2, this implies that (w + x)ρ = (y + z)ρ and (w ⋅ R → S x)ρ = (y ⋅ z)ρ. Further, we have # (ker Φ ) ↓ ↑ α Associative laws R/ker Φ Choose x, y, z ∈R such that xρ, yρ, zρ∈R/ρ, then Proof (i) With respect to ‘+’: We have already proved that R/ρ is a seminear-ring. Now (xρ+ yρ) +zρ = (x + y)ρ+yρ choose x, y ∈R, then

= ((x + y) + z))ρ # # # ρ (x + y) = (x + y)ρ = xρ+yρ = ρ (x) +ρ (y) = ((x + (y + z))ρ = xρ + (y + z)ρ and ρ ρ ρ # # # = x +(y +z ). ρ (x ⋅ y) = (x ⋅ y)ρ = xρ ⋅ yρ = ρ (x) ⋅ ρ (y). (ii) With respect to ‘⋅’: # It follows that ρ is a homomorphism. ker Φ is a (xρ ⋅ yρ) ⋅ yρ = (x ⋅ y)ρ ⋅ zρ congruence relation on R by Theorem 4.1. Now define = ((x ⋅ y) ⋅ z))ρ α : R/ker Φ → S by α (xker Φ ) = Φ (x). Then α = ((x ⋅(y ⋅ z))ρ is well defined as well as one-one, as for all xker Φ , = xρ ⋅(y ⋅ z)ρ yker Φ ∈R/ker Φ = xρ ⋅(yρ ⋅ zρ). xker Φ = yker Φ ⇔ (x, y) ∈ker Φ ⇔ Φ (x) = Φ (y) Thus, R/( ρ, +) and (R/ρ, ) are: semigroups. ⇔ α (xker Φ ) = α (yker Φ ). Right distributive law αis a homomorphism, as for all xkerΦ, yker Φ ∈R/kerΦ Choose x, y, z ∈R such that xρ, yρ, zρ∈R/ρ, then α [(xker Φ ) + (yker Φ )] = α [(x + y)ker Φ ] (yρ+zρ) ⋅ xρ =(y+ z)ρ ⋅ xρ = Φ (x + y) =((y + z) ⋅ x)ρ = Φ (x) Φ (y) = (y ⋅ x + z ⋅ x)ρ = α (xker Φ ) α (yker Φ ) = (y ⋅ x)ρ + (z ⋅ x)ρ and

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α [(xker Φ ) ⋅(yker Φ )] = α [(x ⋅ y)ker Φ ] σ/ρ = {(xρ, yρ) ∈R/ρ×R/ρ: (x, y)∈σ} Φ = (x ⋅ y) is a congruence relation on R/ρ and = Φ (x) Φ (y) R/ρ|σ/ρ ≅ R/σ. = α (xker Φ ) α (yker Φ ). Obviously ranα = ran Φ and from the definition it is Proof clear that for all x∈R First, we show that σ/ρ is a congruence relation.

# Choose r∈R, then (r, r)∈σ, since σ is reflexive. This α (ker Φ ) (x) = α (xker Φ ) = Φ (x). implies that (rρ, rρ)∈σ/ρ, so σ/ρ is reflexive. Now That is, the diagram commutes. choose r, s ∈R such that (rρ, sρ)∈σ/ρ, then (r, s)∈σ and this implies (s, r)∈σ, since σ is symmetric. It follows that (sρ, rρ)∈σ/ρ, so σ/ρ is symmetric. Now choose 4.4 Theorem (Second Isomorphism r, s, t ∈R such that (rρ, sρ), (sρ, tρ)∈σ/ρ, then by defi- Theorem) nition (r, s), (s, t)∈σ and this implies (r, t)∈σ, as σ is Let (R, +, ∙) and (S, , ) be seminear-rings and ρ transitive. It follows that(rρ, tρ)∈σ/ρ, so σ/ρ is transitive. a congruence relation on the seminear-ring R. Assume Thus,σ /ρ is an equivalence relation. that Φ :R → S is a homomorphism such that ρ ⊆ ker Φ Now choose r, s, t, u∈R such that (rρ, sρ), . Then there is a unique homomorphism β : R/ρ → S (tρ, uρ) ∈σ/ρ, then by definition (r, s), (t, u)∈σ . It fol- such that ran β = ran Φ and the diagram given below lows that (r + t, s +u), (r ∙ t, s ∙ u)∈σ, since σ is compatible, commutes. and this implies that Φ ((r + t)ρ, (s +u)ρ), ((r ∙ t)ρ, (s ∙ u)ρ) ∈σ/ρ. R→S It follows that σ/ρis compatible. ρ # ↓ ↑ β Now define a mapping β : R/ρ → R/σ by β (xρ) = xσ. R/ρ Choose xρ, yρ∈R/ρ, then Proof b(xρ+ yρ) = b((x + y)ρ)= (x + y)σ = xσ+yσ= b(xρ ) +b(xρ) Define the mapping β : R/ρ → S by β (xρ) = Φ (x) for all and xρ∈R/ρ. Then β is well-defined, because for allx ρ, yρ∈ b(xρ ∙ yρ) = b((x ∙ y)ρ) = (x ∙ y)σ= xσ ∙ yσ= b(xρ) ∙ b(yρ). R/ρ, xρ = yρ ⇒ (x, y) ∈ρ ⊆ ker Φ ⇒ Φ (x) = Φ (y) ⇒ β (xρ) = β (yρ). It follows that β is a homomorphism. So by Theorem β is a homomorphism, since for xρ, yρ∈R/ρ, 4.3, there is a monomorphism b(xρ+yρ)= b[(x + y)ρ] = Φ (x + y) = Φ(x) ⊕ Φ (y) α : R/ρ|ker β → R/σ = b(xρ) ⊕b(yρ) defined by α ((xρ )ker β ) =x σ. Obviously it is onto. Thus and R/ρ|ker β ≅ R/σ. Now b(xρ ⋅ xρ)= b[(x ⋅ y)ρ] = Φ (x ⋅ y) = Φ (x) Φ (y) ker β = {(xρ, yρ) ∈R/ρ×R/ρ: β (xρ) = β (yρ)} =b(xρ) b(yρ). # = {(xρ, yρ) ∈R/ρ×R/ρ: xσ = yσ } Now β ρ (x)= β (xρ) = Φ (x). It follows that the = {(xρ, yρ) ∈R/ρ×R/ρ: (x, y)∈σ } diagram commutes. Clearly ran β = ran Φ . To show the = σ/ρ. uniqueness, let a: R/ρ → S be another homomorphism # This completes the proof. such that aρ = Φ . Let x∈R, then

# # aρ (x) = Φ(x) = bρ (x) ⇒ a(xρ) = b(xρ) ⇒ a = b. 5. Congruence Relations and c-Ideals

In this section, we show that each c-ideal defines a 4.5 Theorem [Third Isomorphism Theorem] congruence relation on seminear-rings and the idea Let R be a seminear-ring. Assume that ρ and σ are comes from the paper16 in which the author does similar congruence relations on R such that ρ ⊆ σ. Then calculations for semirings.

Vol 9 (38) | October 2016 | www.indjst.org Indian Journal of Science and Technology 5 Quotient Seminear-Rings

5.1 Theorem Associative laws Let I be a c-ideal of a seminear-ring (R, +, ∙). Then, I Choose a, b, c∈R such that a + I, b + I, c + I ∈R/I, then define a congruence relation ρ on R given by (r, s) ρ if (i) With respect to ‘+’: and only if there exist x, y I such that r + x = y + s. (a + I+b + I) +c + I = (a + b) + I+c + I Proof = ((a + b) + c)) + I Let r R and x I. As I is a c-ideals of R, there exists x , 1 = ((a + (b + c)) + I x I such that r + x + x = x + x + r. This implies that 2 1 2 = a + I+ (b + c) + I (r, r) ρ, since I is an ideal. Let (r, s) ρ for some r, s = a + I+(b + I+c + I). R. Then there arex 1, x2 I such that r + x1 = x2 + s. As I is a c-ideal of R, there exist x3, x4, x5, x6 I such that r + (ii) With respect to ‘ ’: x1 + x3 = x4 + x1 + r and s + x2 + x5 = x6 + x2 + s. This implies ∙ ∙ ∙ ∙ x6 + x4+ x1 + r = s + x2 + x5+ x3 which implies that (s, r) (a + I b + I) c + I = (a b) + I c + I ρ. Let r, s, t R such that (r, s) ρ and (s, t) ρ. Then = ((a ∙ b) ∙ c)) + I

there exist x1, x2, y1, y2 I such that r + x1 = x2 + s and = ((a ∙ (b ∙ c)) + I

s + y1 = y2 + t. This implies that r + x1+ y1= x2 + s + y1 = = a + I ∙ (b ∙ c) + I x + y + t which implies (r, t) ρ. It follows that ρ is an 2 2 = a + I ∙ (b+ I ∙ c + I). equivalence relation. Thus R/I( , +) and (R/I, ∙) are semigroups. Again as I is an ideal of R, (r, s) ρ for some r, s R implies (t ∙ r, t ∙ s) ρ, and (r ∙ t,s ∙ t) ρ, for all t R. Right distributive law Finally let r, s, t R such that (r, s) ρ. This implies thatr Choose a, b, c ∈R such that a+ I, b+ I,c+ I∈R/I, then

+ x1 = x2 + s. for some x1, x2 I. Therefore t + (r + x1) + x3 (b+ I+c+ I) ∙ a+ I = (b + c)+ I ∙ a+ I = t + (x2 + s) + x3= x4+ (x2 + s) + t for some x3, x4 I, since I is a c-ideal of R. Thus t( + r,t + s) ρ. Now using (t + r, =((b + c) ∙ a) + I t + s) ρ and by using the fact that I is a c-ideal and ρ is = (b ∙ a + c ∙ a) + I symmetric we may get (r + t, s + t) ρ. Thereforeρ is left as = (b ∙ a) + I+ (c ∙ a) + I well as right congruence relation. This completes the proof. = b+ I ∙ a+ I+c+ I ∙ a+ I Now let ρ be the above congruence relation on the Thus (R/I, +, ∙) is a seminear-ring which is called quo- seminear-ring R. Assume that a+ I represents the con- tient seminear-ring. gruence class corresponding to the element a∈R and R/I represents the set of all congruence classes, i.e. R/I = {a + I : a∈R}. 6. Conclusion The following result shows that R/I becomes a seminear-ring. If we assume that (R, +, ∙) and (S, , ) are two semi-

near-rings with additive identities 0R and 0S, then we say 5.2 Theorem that a map a: R S is a homomorphism if it satisfies Let (R, +, ∙) be a seminear-ring and I c-ideal of R. ThenR/I the conditions of Definition 2.7 along with the condi-

is seminear-ring under the following binary operations: tion a(0R) = 0S. In this case, we may define kernel of the homomorphism a in the usual way. Further if we assume (a + I) + (b + I) = (a + b) + I and (a + I) ∙ (b + I) = (a ∙ b) + I. that x 0S = 0S x =0S for all x S, then we may prove Proof that kernel is an ideal but we don’t know that whether it First we show that the above binary operations are well- is a c-ideal or not. Also we know that intersection of two defined. Choose a, b, c, d∈R such that a + I = c + I and ideals of a seminear-ring T is either empty or an ideal b + I = d + I, then Lemma 4.2, (a, c)∈ρ and (b, d)∈ρ. Thus (a + b, c + d), (a ∙ b, c ∙ d)∈ρ, sinceρ is a congru- of T but we don’t know that whether the intersection of ence relation. Thus again by Lemma 4.2, this implies that two c-ideals is a c-ideal or not. Therefore we are unable (a + b) + I= (c + d)+ I and (a ∙ b) + I = (c ∙ d) + I. Further to prove the analogues of the well known isomorphism we prove theorems.

6 Vol 9 (38) | October 2016 | www.indjst.org Indian Journal of Science and Technology Fawad Hussain, Muhammad Tahir, SaleemAbdullah and Nazia Sadiq

7. References 8. Ihsan J. Seminear-rings characterized by their s-ideals-I. Proc. Japan Acad, Ser, 1995; 71(A): 101-03. 1. Dedekind R. Uber die Theorie der ganzenalgebraiscen 9. Ihsan J. Seminear-rings characterized by their s-ideals-II. Zahlen, Springer Fachmedien Wiesbaden, 1964, p. 297-13. Proc. Japan Acad, Ser, 1995; 71(A):111-13. 2. Vandiver HS. Note on a simple type of algebra in which 10. Shabir M, Ahmed I. Weakly regular seminear-rings. cancellation law of addition does not hold. Bulletin of the International Electronic Journal of Algebra. 2007; 2:114-26. American Mathematical Society. 1934; 40(12):914-20. 11. Zulfiqar M. A note on radicals of seminear-rings. Navi Sad 3. Golan JS. The theory of semirings with applications in J. Math. 2009; 39(1):65-8. mathematics and theoretical computer science, John Willy 12. Perumal R, Balakrishnan R. Left bipotentseminear-rings. International Journal of Algebra. 2012; 6(26):1289-95. and Sons Inc.: New York, NY, 1992. 13. Balakrishnan R, Perumal R. Left duo seminear-rings. 4. Praba B, Chandrasekaran VM, Manimaran A. Semiring on Scientia Magna. 2012; 8(3):115-20. Roughsets. Indian Journal of Science and Technology. 2015 14. Kazemipour SA, Fozouni M. Quotient Amenability of Feb; 8(3):1-7. Banach Algebras. Indian Journal of Science and Technology. 5. Anjaneyulu GSGN, Sanyasirao A. Distributed Group Key 2015 Jul; 8(13):1-7. Management Protocol over Non-commutative Division 15. Moghaddam MA. f-best Coapproximation in Quotient Semirings. Indian Journal of Science and Technology. 2014 Topological Vector Spaces. Indian Journal of Science and Jan; 7(6):1-6. Technology. 2014 Jan; 7(12):1-4. 6. Hoorn WG, Rootoselaar B. Fundamental notions in the 16. Shamik G. A characterization of ring congruences on theory of seminear-rings. Comp. Mathematica. 1967; 18(1- semirings. Soochow J. Math. 1993; 19(3):305-32. 2):65-78. 17. Howie JM. Fundamental of semigroup theory, Clarendon 7. Weinert HJ. Seminear-rings, seminear-fields and their Press Oxford: USA, 1995. semigroup theoretical background. Semigroup Forum. 18. Pinter C. Set theory, Addison-Wesley Publishing Co; 1982; 24(1):231-54. London, 1971.

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