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JNTS No 47 / 2018 (XXIII) 1 JNTS No 47 / 2018 (XXIII) PUBLISHED BY THE ACADEMY OF SCIENCES OF ALBANIA JNTS JOURNAL OF NATURAL AND TECHNICAL SCIENCES 2018 (2) XXIII (47) 2 JNTS No 47 / 2018 (XXIII) 3 JNTS No 47 / 2018 (XXIII) ACKNOWLEDGEMENT TO REVIEWERS AND EDITORS The Editor-in-Chief is very thankful to his Editorial Board team for their continued support and diligent, hard work. But I am especially thankful to our large groups of very faithful and excellent reviewers without whom we wouldn’t have a journal of quality. Every paper is reviewed by a minimum of two reviewers and I expect authors of accepted papers to reciprocate by writing really good reviewed version of their manuscripts as well. Reviewing papers is one of the best ways to stay on top of developments in your area of research and it is a necessary component of your career path in Academia. Reviewing is a hard and time demanding process. Most of the time, for better or worse, we respect the final decisions of the reviewers. In the case of very conflicting reviews, we call for a third reviewer which, of course, prolongs the review process. Thank you for the time spent and constructive comments on manuscripts to the following reviewers’ team: Abedin Xhomo Ilia Mikerezi Alaudin Kodra Illir Alliu Alda Kika Indrit Baholli Alfred Frashëri Indrit Inesi Angelo Desantis Irakli Prifti Arjan Harxhi Isak Shabani Asti Papa Jul Bushati Blerina Kolgjini Klodian Zaimi Donika Prifti Kostandin Dollani Dorian Minarolli Luljeta Bozo Enea Mustafaraj Lulzim Hanelli Flora Qarri Matilda Mema Frederik Dara Mentor Lame Gafurr Muka Nevton Kodhelaj Genc Sulcebe Niko Pano Gjergji Foto Pëllumb Berberi Ibrahim Milushi Pëllumb Sadushi Ilda Kazani Rigena Sema 4 JNTS No 47 / 2018 (XXIII) Rustem Paci Steljan Buzo Shaqir Nazo Theodor Karaj Shyqyri Aliaj Vilson Bare Spiro Thodhorjani Vilson Silo With Best Regards from the Editor, the whole Editorial Board and myself, Yours sincerely, Salvatore Bushati Editor-in-Chief 5 JNTS No 47 / 2018 (XXIII) GREEN’S RELATION B IN RINGS AND IN THEIR MULTIPLICATIVE SEMIGROUPS Florion ÇELA and Petraq PETRO Department of Mathematics, Faculty of Natural Sciences, University of Tirana, Albania ____________________________________________________________ ABSTRACT Green’s relations in semigroups are successful tools for studying properties of semigroups. These relations areintroduced and studied also in rings. Gradually they have been transformed into tools not only for studying rings, but also to point out the connection between rings and their multiplicative semigroups. Firstly, in this paper we give solution to an open problem that has to do with equivalence classes of Green’s relation in rings. Further we find some conditions that show when the Green’s relation in ring is equal to Green’s relation (.) in the multiplicative semigroup of this ring. Lastly, we point out the rings in which the Green’s relations and (.) coincide. These rings are very close to regular rings, but we show by giving a counterexample, that they differ from each other. Keywords: semigroups, rings, regular rings, Green’s relations, bi-ideals. 1. INTRODUCTION In this paper by a ring we mean an associative ring, which does not necessary have an identity element. The relations and are first introduced and studied by Petro (1981; 2002; 2014) and further by Sema (2014). These relations are called Green’s relations in rings because they mimic the Green’s relations and in semigroups, which are first introduced and studied by Green (1951). In a similar way like Green’s relations in rings, in (Sema 2014) was introduced and studied another relation which is denoted by and called Green’s relation . In fact, the Green’s relation in rings mimics the relation in semigroups which was first introduced and studied by Kupp (1969) which we will call it Green’s relation in semigroup. First, we aim to give solution to an open problem raised in 6 JNTS No 47 / 2018 (XXIII) (Sema, 2014). For this, we study the Green’s relation in a ring and the Green’s relation (.) in the multiplicative semigroup of this ring, also. Further, we find some sufficient conditions and also a necessary and sufficient condition that show when the Green’ s relation ina ring coincide to Green’s relation (.) in the multiplicative semigroup of this ring. Hence, we find some conclusions when the relation coincides with the relation (.) and when they differ from each other. Lastly, after pointing out that in the class of regular rings the relation coincide with the relation (.) we will show that the class of rings in which the relation and (.) coincide and which we also don’t have equivalence classes with more than one element, respect to relation coincide with the class of regular rings. By a counter example we show that, even though the class of the rings in which the = (.) is very close to regular rings, they differ from each other. In the end, we raise an open problem that has to do with a characterization of the class of regular rings by the equivalence classes of the Green’s relation . 2. PRELIMINARIES We give some notions and present some auxiliary result that will be used throughout the paper. For all unexplained concepts and propositions the reader may refer to (Steinfeld, 1978) and (Howie, 1995). Let (A , ,.) be a ring and BC, two subsets of A . We write: B C={ b c A | b B , c C }, n BC= { b c A | b B , c C }. i=1 i i i For simplicity we will write bC, bC, Bc instead of {}bC, {}bC and Bc{}. A subring B of a ring (,,)A is called left (right) bi-ideal BRL(,) if the following inclusion is satisfied BAB B (,)RA R LA L (Steinfeld, 1978). Green’s relations ,,in a ring A are defined as below(Petro 1981; 2002): (a , b ) A2 , a b ( a ) = ( b ) rr, (a , b ) A2 , a b ( a ) = ( b ) ll, (a , c ) A2 , a b ( a ) = ( c ) bb, 7 JNTS No 47 / 2018 (XXIII) where (),(),abrr((a ),(l b ),( l a ),()) b c b are respectively principle right ideals, principal left ideals, principal bi-idealswhich are generated by the elements abc,, . It is not difficult to prove that for each elementa of a ring, the following equalities are satisfied: (a)r=ℤa+Aa, (1.1) (a)r=ℤa+aA, (1.2) 2 (a)r=ℤa+ℤa +aAa. (1.3) The relations , and in the ring are equivalence relations in its support. The equivalence classes for an element aAmodℛ, modℒ, modℬ are denoted respectively a,,. a a The subsemigroup B of a semigroup (S,.)B is called bi-ideal of this semigroup (S,.) if BSB⊆B. Definition 2.1. (Kapp, 1969) Let (,)S be a semigroup. For each two elements ab, of S we write ac if ac= or there exist elements uv, of S such that: a= buc , c= ava . Proposition 2.2. (Mielke, 1969) Let be a semigroup. Then 2 (,),a b S a c( a )bb = ( c ) , where ()a b and ()c b are principle bi-ideals (the intersection of all bi- ideals that contain respectively elements a and c ) generated by elements ac, of S . It is clear thatthe relation(,,)A in a semigroup is an equivalence relation on S which we will call it Green’s relation ℬ. If in the above definitions of the Green’s relations , and in a ring we replace the ring with the semigroup , by using the Proposition 2.2 and by regarding the principal right ideals, principal left ideals and principal bi-ideals of the ring (A,+,.) respectively as principal right ideals, principal left ideals and principal bi-ideals of the semigroup (S,.), then we have the respective Green’s relation’s in the semigroup . Throughout this paper, Green’s relationsin the multiplicative semigroup (,)A of the ring (A , ,.) are denoted (.), (.), (.) in order to distinguish them from 8 JNTS No 47 / 2018 (XXIII) Green’s relations ,, in the ring(A , ,.) . For the same purpose the equivalence classes of an element a withrespect to Green’ s relations in the multiplicative semigroup of the ring (A , ,.) are denoted RLBa(.), a (.), a (.) . Regarding -classes of the ring (A , ,.) and (.) -classes of the multiplicative semigroup of this ring in (Sema, 2014) is raised the following problem: Problem 2.3. In a ring (A , ,.) for an element aA is it true that a is a union of a (.) classes of the multiplicative semigroup (A ,.) of the ring (A , ,.) which have only one element or satisfy the equality BBaa= (.) ? Proposition 2.4. (Howie, 1995) Let S be a cancellable semigroup (that is a semigroup in which for all abc,, ): (ca = ca ) ( a = b ), (ac = bc ) ( a = b ), and suppose that S has not an identity , then: = = 1S , where , are Green’s relations in semigroup S and 1S is the identic relacion in S . 3. MAIN RESULT If the ring (A , ,.) has an identity then from equality (1.3) is evident (.) . In general, by using equality (1.3) we get that aa(.) and consequently BB(.) . By an example (Sema, 2014) is shown that these inclusion is strict. We give a positive answer to Problem 2.3 by the following theorem. Theorem 3.1.Let (A , ,.) be a ring and a any element of A . Then - classes a is either a union of the (.) classes of the multiplicative semigroup (A ,.) of the ring (A , ,.) , which must have a single element or BBaa= (.) , Proof. Let us consider the two following cases: Case 1. Each class Bx (.) that is included in a has only one element. Thus we have BB= (.) . a xa x (,)A 9 JNTS No 47 / 2018 (XXIII) Case 2. There exists a class Bx (.), xBa , such that has at least two elements.
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