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SOME PROPERTIES OF (m, n)-POTENT CONDITIONS
SOME PROPERTIES OF (m, n)-POTENT CONDITIONS
Journal of Applied Mathematics & Informatics. 2015. May, 33(3_4): 469-474
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : June 22, 2014
  • Accepted : July 23, 2014
  • Published : May 30, 2015
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YONG UK CHO

Abstract
In this paper, we will consider the notions of ( m , n )-potent conditions in near-rings, in particular, a near-ring R with left bipotent or right bipotent condition. We will derive some properties of near-rings with (1, n ) and ( n , 1)-potent conditions where n is a positive integer, and then some properties of near-rings with ( m , n )-potent conditions. Also, we may discuss the behavior of R -subgroups in (1, n )-potent or ( n , 1)-potent near-rings.. Mathematics Subject Classification : 16Y30.
Keywords
1. Introduction
The concept of Von Neumann regularity of near-rings have been studied by many authors Beidleman [2] , Choudhari [3] , Heatherly [4] , Ligh, Mason [5] , Murty, and Szeto [9] . Their main results are appeared in the book of Pilz [8] .
In 1980, Mason [5] introduced the notions of left regularity, right regularity and strong regularity of near-rings.
In 1985, Ohori [7] investigated the characterization of π -regularity and strong π -regularity of rings.
A near-ring R is an algebraic system ( R , +, ·) with two binary operations + and · such that ( R , +) is a group (not necessarily abelian) with a zero element 0, ( R , ·) is a semigroup and ( a + b ) c = ac + bc for all a , b , c in R .
A near-field is a unitary near-ring ( F , +, ·) where ( F = F ╲ {0}, ·) is a group [8] .
A near-ring R with the extra axiom a 0 = 0 for all a R is said to be zero symmetric . An element d in R is called distributive if d ( a + b ) = da + db for all a and b in R .
We will use the following notations. Given a near-ring R , R 0 ={ a R | a 0 = 0} which is called the zero symmetric part of R , Rc ={ a R | a 0 = a } which is called the constant part of R . The set of all distributive elements in R is denoted by Rd .
In 1979, Jat and Choudhari defined a near-ring R to be left bipotent (resp. right bipotent) if Ra = Ra 2 (resp. aR = a 2 R ) for each a in R . Also, we can define a near-ring R as subcommutative if aR = Ra for all a in R like as in ring theory. Obviously, every commutative near-ring is subcommutative. From these above two concepts it is natural to investigate the near-ring R with the properties aR = Ra 2 (resp. a 2 R = Ra ) for each a in R . We say that such is a near-ring with (1, 2)-potent conditions (resp. a near-ring with (2, 1)-potent conditions). Thus, from this motivation, we can extend a general concept of a near-ring R with ( m , n )-potent conditions.
First, we will derive properties of near-ring with (1, 2) and (2, 1)-potent conditions, also (1, n ) and ( n , 1)-potent conditions where n is a positive integer. Any homomorphic image of ( m , n )-potent near-ring is also ( m , n )-potent.
Next, we will find some properties of regular near-rings with ( m , n )-potent conditions. Also, we will discuss the behavior of R -subgroups in (1, n )-potent or ( n , 1)-potent near-rings.
For the rest of basic concepts and results on near-rings, we will refer to [8] .
2. Results on (m,n)-potent conditions in Near-Rings
Let R and S be two near-rings. Then a mapping f from R to S is called a near-ring homomorphism if (i) f ( a + b ) = f ( a ) + f ( b ), (ii) f ( ab ) = f ( a ) f ( b ). We can replace homomorphism by monomorphism, epimorphism, isomorphism, endomorphism and automorphism as in ring theory [1] .
We say that a near-ring R has the insertion of factors property (briey, IFP) provided that for all a , b , x in R with ab = 0 implies axb = 0. A near-ring R is called reversible if for any a , b R , ab = 0 implies ba = 0. On the other hand, we say that R has the reversible IFP in case R has the IFP and is reversible.
Also, we say that R is reduced if R has no nonzero nilpotent elements, that is, for each a in R , an = 0, for some positive integer n implies a = 0. McCoy [6] proved that R is reduced iff for each a in R , a 2 = 0 implies a = 0.
A (two-sided) R-subgroup of R is a subset H of R such that (i) ( H , +) is a subgroup of ( R , +), (ii) RH H and (iii) HR H . If H satisfies (i) and (ii) then it is called a left R-subgroup of R . If H satisfies (i) and (iii) then it is called a right R-subgroup of R .
Let ( G , +) be a group (not necessarily abelian) with the identity element o . In the set
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of all the self maps of G , if we define the sum f + g of any two mappings f , g in M ( G ) by the rule ( f + g ) x = fx + gx for all x G and the product f · g by the rule ( f · g ) x = f ( gx ) for all x G , here, for convenience we write the image of f at a variable x , fx instead of f ( x ), then ( M ( G ),+, ·) becomes a near-ring. It is called the self map near-ring on the group G . Also, if we can define the set
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then ( M 0 ( G ),+, ·) is a zero symmetric near-ring, and
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then ( M 0 ( G ),+, ·) is a constant near-ring. ( G , +) is abelian if and only if ( M ( G ), +) is abelian.
A near-ring R [8] is called simple if it has no non-trivial ideal, that is, R has no ideals except 0 and R . Also, R is called R-simple if R has no R -subgroups except R 0 and R .
A near-ring R is called left regular (resp. right regular) if for each a in R , there exists an element x in R such that
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A near-ring R is called strongly left regular if R is left regular and regular, similarly, we can define strongly right regular. A strongly left regular and strongly right regular near-ring is called strongly regular near-ring .
A near-ring R is called left κ-regular (resp. right κ-regular) if for each a in R , there exists an element x in R such that
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for some positive integer n . A left κ -regular and right κ -regular near-ring is called κ-regular near-ring .
An integer group (ℤ 2 , +) modulo 2 with the multiplication rule. 0·0 = 0·1 = 0, 1·0 =1·1 =1 is a near-field. Obviously, this near-field is isomorphic to Mc (ℤ 2 ). All other near-fields are zero-symmetric. Consequently, we get the following important statement.
Lemma 2.1 ( [8] ). Let R be a near-field. Then R Mc (ℤ 2 ) or R is zero-symmetric .
In our subsequent discussion of near-fields, we will exclude the silly near-field Mc (ℤ 2 ) of order 2. Evidently, every near-field is simple.
Lemma 2.2 ( [8] ). Let R be a near-ring. Then the following statements are equivalent .
(1) R is a near-field .
(2) Rd ≠ 0 and for each nonzero element a in R, Ra = R .
(3) R has a left identity and R is R-simple .
From now on, we give the new concept of an ( m , n )-potent near-ring, and then illustrate this notion with suitable examples.
We say that a near-ring R has the ( m , n )- potent condition if for all a in R , there exist positive integers m , n such that amR = Ran . We shall refer to such a near-ring as an ( m , n )- potent near-ring .
Obviously, every ( m , n )-potent near-ring is zero-symmetric. On the other hand, from the Lemmas 2.1 and 2.2, we obtain the following examples (1), (2).
Examples 2.3. (1) Every near-field is an ( m , n )-potent near-ring for all positive integers m , n .
  • (2) The direct sum of near-fields is an (m,n)-potent near-ring for all positive integersm,n.
  • (3) Every subcommutative near-ring is an (1, 1)-potent near-ring.
  • (4) Every Boolean subcommutative near-ring is an (m,n)-potent near-ring for all positive integersm,n.
  • (5) LetR={0,a,b,c} be an additive Klein 4-group. This is a near-ring with the following multiplication table (p. 408[8]).
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  • This near-ring have several (1, 1), (1, 4), (2, 2), (2, 4), (3, 3), (4, 1), (4, 2) and (4, 4)-potent conditions, but it is not a Boolean near-ring.
  • A near-ringRis calledleft S-unital (resp. right S-unital)if for eachainR,a∈Ra(resp.a∈aR).
Lemma 2.4. Let R be a zero-symmetric and reduced near-ring. Then R has the reversible IFP .
Proof . Suppose that a , b in R such that ab = 0. Then, since R is zero-symmetric, we have
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Reducedness implies that ba = 0.
Next, assume that for all a , b , x in R with ab = 0. Then
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This implies axb = 0, by reducedness. Hence R has the reversible IFP. □
Theorem 2.5. Let R be an ( n , n + 2)- potent reduced near-ring, for some positive integer n. Then is a left κ-regular near-ring .
Proof . Suppose R is an ( n , n + 2)-potent reduced near-ring. Then for any a in R , we have that
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This implies that a n+1 anR = Ra n+2 . Hence there exists x in R such that a n+1 = xa n+2 , that is, ( an xa n+1 ) a = 0. From Lemma 2.4, we see that a ( an xa n+1 ) = 0. Also, we can compute that an ( an xa n+1 ) = 0 and xa n+1 ( an xa n+1 ) = 0. Thus from the equation
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and reducedness, we see that an = xa n+1 . Consequently, R is a left κ -regular near-ring. □
Corollary 2.6. Let R be an (1, 3)- potent reduced near-ring. Then R is a left regular near-ring .
Theorem 2.7. Let R be an (1, 2)- potent near-ring. Then we have the following statements.
  • (1)If R is reduced, then R is a left S-unital near-ring.
  • (2)If R is right S-unital, then R is a left regular and reduced near-ring.
Proof . Since R is an (1, 2)-potent near-ring, consider the equality, aR = Ra 2 for each a in R .
  • (1) Froma2∈aR=Ra2, there existsxinRsuch thata2=xa2. This implies that (a−xa)a= 0. SinceRis zero-symmetric and reduced, Lemma 2.4 guarantees thata(a−xa) = 0 andxa(a−xa) = 0. Hence we have the equation
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  • Reducedness implies that (a−xa) = 0, that is,a=xa, for somex∈R. ThereforeRis left S-unital.
  • (2) SinceRis right S-unital and has (1, 2)-potent condition, for eacha∈R,a∈aR=Ra2. Thusa=xa2, for somex∈R. Also, in this equation,a2= 0 implies thata= 0. HenceRis a left regular and reduced near-ring. □
Proposition 2.8. Let R be an (2, 1)- potent near-ring. Then we have the fol- lowing statements .
  • (1)If R = Rdis reduced, then R is a right S-unital near-ring.
  • (2)If R is left S-unital, then R is a right regular and reduced near-ring.
Proof . This proof is an analogue of the proof in Proposition 2.7. □
Theorem 2.9. Every homomorphic image of an ( m , n )- potent near-ring is also an ( m , n )- potent near-ring .
Proof . Let R be an ( m , n )-potent near-ring and let f . R R′ be a near-ring epimorphism. Consider an equality amR = Ran , for all a R , where m , n are positive integers.
We must show that for all a′ R′ , a′mR′ = R′ a′n , for some positive integers m , n . Let a′ , x′ R′ . Then there exist a , x R such that a′ = f ( a ) and x′ = f ( x ). So we get the following equations.
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where amx amR = Ran , so that there exist y R such that amx = yan . This implies that a′ mR′ R′a′ n .
In a similar fashion, we obtain that R′a′ n a′ mR′ . Therefore our desired result is completed. □
Finally, we may discuss the behavior of R -subgroups of (1, n )-potent near-ring as following.
Proposition 2.10. Every left R-subgroup of an (1, n )- potent near-ring R is an R-subgroup .
Proof . Let A be a left R -subgroup of R . Then we see that RA A . To show that AR A , let ar AR , where a A , r R . Since R has (1, n )-potent condition, we have ar aR = Ran . This implies that
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for some s in R . Hence A is an R -subgroup of R . □
BIO
Y.U. Cho visited five times at Univ. in U.S.A and Japan as a visiting professor, one time, the Univ. of Louisiana at Lafayette in U.S.A (one year 1996.9-1997.8), second time, the Ohio Univ. at Athens in U.S.A (one year 2002.3-2003.3), two times, the Okayama Univ. in Japan (1999 one month, 2004 one month, during summer vacations), fifth time, the Florida Atlantic Univ. at Florida in U.S.A (one year 2008.8-2009.7).
Now he is acting members of the editorial board of JP-Journal of Algebra and Number Theory (:JPANT) from 2000 until now, East Asian Math Journal from 1999 until 2006, and American Journal of Applied Mathematics and Mathematical Sciences (:AJAMMS) from 2010 till now. Also, he is an Associate Editor of JAMI from 2008 till now. His research interests focus on the theory of near-rings.
Department of Mathematics Education, College of Education, Silla University, Pusan 617-736, Korea
e-mail: yucho@silla.ac.kr
References
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Choudhari S.C. , Jat J.L. (1979) On left bipotent near-rings Proc. Edin. Math. Soc. 22 99 - 107    DOI : 10.1017/S0013091500016217
Heatherly H.E. (1974) On regular near-rings J. Indian Math. Soc. 38 345 - 354
Mason G. (1980) On strongly regular near-rings Proc. Edn. Math. Soc. 27 - 36    DOI : 10.1017/S0013091500003564
McCoy N.H. (1969) The theory of rings Macmillan and Cc
Ohori M. (1985) On stronglyπ-regular rings and periodic rings Math. J. Okayama Univ. 27 49 - 52
Pilz G. (1983) Near-rings North Holland Publishing Company Amsterdam, New York, Oxford
Szeto G. (1974) On regular near-rings with no nonzero nilpotent elements Math. Japon. 79 65 - 70