In this paper, we will consider the notions of (
m
,
n
)potent conditions in nearrings, in particular, a nearring
R
with left bipotent or right bipotent condition. We will derive some properties of nearrings with (1,
n
) and (
n
, 1)potent conditions where
n
is a positive integer, and then some properties of nearrings with (
m
,
n
)potent conditions. Also, we may discuss the behavior of
R
subgroups in (1,
n
)potent or (
n
, 1)potent nearrings..
Mathematics Subject Classification : 16Y30.
1. Introduction
The concept of Von Neumann regularity of nearrings 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 nearrings.
In 1985, Ohori
[7]
investigated the characterization of
π
regularity and strong
π
regularity of rings.
A
nearring 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
nearfield
is a unitary nearring (
F
, +, ·) where (
F
^{∗}
=
F
╲ {0}, ·) is a group
[8]
.
A nearring
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 nearring
R
,
R
_{0}
={
a
∈
R

a
0 = 0} which is called the
zero symmetric part
of
R
,
R_{c}
={
a
∈
R

a
0 =
a
} which is called the
constant part
of
R
. The set of all distributive elements in
R
is denoted by
R_{d}
.
In 1979, Jat and Choudhari defined a nearring
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 nearring
R
as subcommutative if
aR
=
Ra
for all
a
in
R
like as in ring theory. Obviously, every commutative nearring is subcommutative. From these above two concepts it is natural to investigate the nearring
R
with the properties
aR
=
Ra
^{2}
(resp.
a
^{2}
R
=
Ra
) for each
a
in
R
. We say that such is a nearring with (1, 2)potent conditions (resp. a nearring with (2, 1)potent conditions). Thus, from this motivation, we can extend a general concept of a nearring
R
with (
m
,
n
)potent conditions.
First, we will derive properties of nearring 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 nearring is also (
m
,
n
)potent.
Next, we will find some properties of regular nearrings with (
m
,
n
)potent conditions. Also, we will discuss the behavior of
R
subgroups in (1,
n
)potent or (
n
, 1)potent nearrings.
For the rest of basic concepts and results on nearrings, we will refer to
[8]
.
2. Results on (m,n)potent conditions in NearRings
Let
R
and
S
be two nearrings. Then a mapping
f
from
R
to
S
is called a
nearring 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 nearring
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 nearring
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
,
a^{n}
= 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 (twosided)
Rsubgroup
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 Rsubgroup
of
R
. If
H
satisfies (i) and (iii) then it is called a
right Rsubgroup
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 nearring. It is called the
self map nearring on the group G
. Also, if we can define the set
then (
M
_{0}
(
G
),+, ·) is a zero symmetric nearring, and
then (
M
_{0}
(
G
),+, ·) is a constant nearring. (
G
, +) is abelian if and only if (
M
(
G
), +) is abelian.
A nearring
R
[8]
is called
simple
if it has no nontrivial ideal, that is,
R
has no ideals except 0 and
R
. Also,
R
is called
Rsimple
if
R
has no
R
subgroups except
R
_{0}
and
R
.
A nearring
R
is called
left regular (resp. right regular)
if for each
a
in
R
, there exists an element
x
in
R
such that
A nearring
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 nearring is called
strongly regular nearring
.
A nearring
R
is called
left κregular (resp. right κregular)
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 nearring is called
κregular nearring
.
An integer group (ℤ
_{2}
, +) modulo 2 with the multiplication rule. 0·0 = 0·1 = 0, 1·0 =1·1 =1 is a nearfield. Obviously, this nearfield is isomorphic to
M_{c}
(ℤ
_{2}
). All other nearfields are zerosymmetric. Consequently, we get the following important statement.
Lemma 2.1
(
[8]
).
Let R be a nearfield. Then R
≅
M_{c}
(ℤ
_{2}
)
or R is zerosymmetric
.
In our subsequent discussion of nearfields, we will exclude the silly nearfield
M_{c}
(ℤ
_{2}
) of order 2. Evidently, every nearfield is simple.
Lemma 2.2
(
[8]
).
Let R be a nearring. Then the following statements are equivalent
.
(1)
R is a nearfield
.
(2)
R_{d}
≠ 0
and for each nonzero element a in R, Ra
=
R
.
(3)
R has a left identity and R is Rsimple
.
From now on, we give the new concept of an (
m
,
n
)potent nearring, and then illustrate this notion with suitable examples.
We say that a nearring
R
has the (
m
,
n
)
potent condition
if for all
a
in
R
, there exist positive integers
m
,
n
such that
a^{m}R
=
Ra^{n}
. We shall refer to such a nearring as an (
m
,
n
)
potent nearring
.
Obviously, every (
m
,
n
)potent nearring is zerosymmetric. On the other hand, from the Lemmas 2.1 and 2.2, we obtain the following examples (1), (2).
Examples 2.3.
(1) Every nearfield is an (
m
,
n
)potent nearring for all positive integers
m
,
n
.

(2) The direct sum of nearfields is an (m,n)potent nearring for all positive integersm,n.

(3) Every subcommutative nearring is an (1, 1)potent nearring.

(4) Every Boolean subcommutative nearring is an (m,n)potent nearring for all positive integersm,n.

(5) LetR={0,a,b,c} be an additive Klein 4group. This is a nearring with the following multiplication table (p. 408[8]).

This nearring 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 nearring.

A nearringRis calledleft Sunital (resp. right Sunital)if for eachainR,a∈Ra(resp.a∈aR).
Lemma 2.4.
Let R be a zerosymmetric and reduced nearring. Then R has the reversible IFP
.
Proof
. Suppose that
a
,
b
in
R
such that
ab
= 0. Then, since
R
is zerosymmetric, 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 nearring, for some positive integer n. Then is a left κregular nearring
.
Proof
. Suppose
R
is an (
n
,
n
+ 2)potent reduced nearring. Then for any
a
in
R
, we have that
This implies that
a
^{n+1}
∈
a^{n}R
=
Ra
^{n+2}
. Hence there exists
x
in
R
such that
a
^{n+1}
=
xa
^{n+2}
, that is, (
a^{n}
−
xa
^{n+1}
)
a
= 0. From Lemma 2.4, we see that
a
(
a^{n}
−
xa
^{n+1}
) = 0. Also, we can compute that
a^{n}
(
a^{n}
−
xa
^{n+1}
) = 0 and
xa
^{n+1}
(
a^{n}
−
xa
^{n+1}
) = 0. Thus from the equation
and reducedness, we see that
a^{n}
=
xa
^{n+1}
. Consequently,
R
is a left
κ
regular nearring. □
Corollary 2.6.
Let R be an
(1, 3)
potent reduced nearring. Then R is a left regular nearring
.
Theorem 2.7.
Let R be an
(1, 2)
potent nearring. Then we have the following statements.

(1)If R is reduced, then R is a left Sunital nearring.

(2)If R is right Sunital, then R is a left regular and reduced nearring.
Proof
. Since
R
is an (1, 2)potent nearring, 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 zerosymmetric and reduced, Lemma 2.4 guarantees thata(a−xa) = 0 andxa(a−xa) = 0. Hence we have the equation

Reducedness implies that (a−xa) = 0, that is,a=xa, for somex∈R. ThereforeRis left Sunital.

(2) SinceRis right Sunital 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 nearring. □
Proposition 2.8.
Let R be an
(2, 1)
potent nearring. Then we have the fol lowing statements
.

(1)If R = Rdis reduced, then R is a right Sunital nearring.

(2)If R is left Sunital, then R is a right regular and reduced nearring.
Proof
. This proof is an analogue of the proof in Proposition 2.7. □
Theorem 2.9.
Every homomorphic image of an
(
m
,
n
)
potent nearring is also an
(
m
,
n
)
potent nearring
.
Proof
. Let
R
be an (
m
,
n
)potent nearring and let
f
.
R
→
R′
be a nearring epimorphism. Consider an equality
a^{m}R
=
Ra^{n}
, for all
a
∈
R
, where
m
,
n
are positive integers.
We must show that for all
a′
∈
R′
,
a′^{m}R′
=
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
a^{m}x
∈
a^{m}R
=
Ra^{n}
, so that there exist
y
∈
R
such that
a^{m}x
=
ya^{n}
. This implies that
a′ ^{m}R′
⊂
R′a′ ^{n}
.
In a similar fashion, we obtain that
R′a′ ^{n}
⊂
a′ ^{m}R′
. Therefore our desired result is completed. □
Finally, we may discuss the behavior of
R
subgroups of (1,
n
)potent nearring as following.
Proposition 2.10.
Every left Rsubgroup of an
(1,
n
)
potent nearring R is an Rsubgroup
.
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
=
Ra^{n}
. This implies that
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 1996.91997.8), second time, the Ohio Univ. at Athens in U.S.A (one year 2002.32003.3), 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 2008.82009.7).
Now he is acting members of the editorial board of JPJournal 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 nearrings.
Department of Mathematics Education, College of Education, Silla University, Pusan 617736, Korea
email: yucho@silla.ac.kr
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