SOME PROPERTIES OF (m, n)-POTENT CONDITIONS

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..

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 a0 = 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, R0 ={a ∈ R | a0 = 0} which is called the zero symmetric part of R, Rc ={a ∈ R | a0 =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 =Ra2 (resp. aR =a2R) for each a in R. Also, we can define a near-ring R as subcommutative if aR =Rafor 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 =Ra2 (resp. a2R =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, a2 = 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

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

then (M0(G),+, ·) is a zero symmetric near-ring, and

then (M0(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 R0 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

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

for some positive integer n. A left κ-regular and right κ-regular near-ring is called κ-regular near-ring.

An integer group (ℤ2, +) 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(ℤ2). 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(ℤ2) or R is zero-symmetric.

In our subsequent discussion of near-fields, we will exclude the silly near-field Mc(ℤ2) 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.

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

Reducedness implies that ba = 0.

Next, assume that for all a, b, x in R with ab = 0. Then

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

This implies that an+1 ∈ anR = Ran+2. Hence there exists x in R such that an+1 =xan+2, that is, (an −xan+1)a = 0. From Lemma 2.4, we see that a(an−xan+1) = 0. Also, we can compute that an(an − xan+1) = 0 and xan+1(an − xan+1) = 0. Thus from the equation

and reducedness, we see that an =xan+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.

Proof. Since R is an (1, 2)-potent near-ring, consider the equality, aR = Ra2 for each a in R.

Proposition 2.8. Let R be an (2, 1)-potent near-ring. Then we have the fol- lowing statements.

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.

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

for some s in R. Hence A is an R-subgroup of R. □