A generalisation of regularities in near-rings

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A generalisation of regularities in near-ringsN.J. Groenewald a & P.C. Potgieter aa Department of Mathematics , University of Port Elizabeth , P.O.Box 1600,Port Elizabeth, 6000, South AfricaPublished online: 27 Jun 2007.

To cite this article: N.J. Groenewald & P.C. Potgieter (1989) A generalisation of regularities in near-rings,Communications in Algebra, 17:6, 1449-1462, DOI: 10.1080/00927878908823799

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COMMUNICATIONS IN ALGEBRA, 1 7 ( 6 ) , 1449-1462 (1989)

A GENERALISATION OF REGULARITIES I N NEAR- RINGS

N.J. Groenewald and P C. Po tg ie te r

Department of Mathematics Univers i ty of Por t El izabeth

P.O.Box 1600 Por t El izabeth

6000 South Af r ica

Cer ta in r e g u l a r i t i e s i n a s s o c i a t i v e r i n g s have been known f o r a

long time. Most of these r e g u l a r i t i e s have s ince been def ined

f o r near- r i n g s , a s i s evident from t h e examples i n paragraph 4.

I n t h i s paper we s h a l l attempt t o develop a genera l theory of

r e g u l a r i t i e s f o r near- r ings . Boos managed t o de f ine t h e concept

of a r e g u l a r i t y f o r r i n g s i n genera l i n h i s Ph.D. d i s s e r t a t i o n

[lo]. The r e s u l t s i n t h i s a r t i c l e is based, i n p a r t , on t h e work

done by him i n t h i s context .

1. BASIC CONCEPTS

Throughout t h i s paper we u t i l i z e t h e no ta t ions and d e f i n i t i o n s

from P i l z [7] .

Copyright O 1989 by Marcel Dekker, Inc.

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1450 GROENEWALD AND POTGIETER

By a n e a r - r i n g we sha l l mean an algebraic system (N,+,-) where

( i ) (N,+) forms a group,

( i i ) (N, . ) forms a semi- group and

( i i i ) (a+b)c = ab+ac f o r a l l a ,b ,c E N [ i .e . we work with r ight

near- rings] .

A normal subgroup I of N i s called a r i g h t i d e a l [I a T N] i f

IN g I. It i s called a l e f t i d e a l [I q1 N] i f n (n t+ i ) - nn) E I

fo r a l l n,nl E N and a l l i E I . I f such a normal subgroup I i s

both a l e f t and a r ight ideal in N , it i s called an ideal in N

[I a N] . Homomorphisms, isomorphisms, e t c . fo r near- rings are

defined i n the usual way (cf . [7] ) . Ideals in a near- ring w i l l

thus be kernels of homomorphisms . We sha l l use the following

characterization of radical classes. A subclass R of a universal

class W of near-rings i s a Kurosh- Amitsur radical class

[KA- radical class] i f it s a t i s f i e s the following properties:

K1. R i s homomorphically closed.

K 2 . Any near-ring N contains an ideal W(N) from !R tha t contains

a l l ideals from lR in N .

K3. iR(N/R(N)) = 0 fo r any near-ring N .

2 . F- REGULARITY

With every near-ring N we associate a mapping FN : N + S(N,+)

where S(N,+) i s the se t of a l l subgroups of the additive group

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A GENERALISATION OF REGULARITIES IN NEAR-RINGS 1451

(N,+) of N . Le t T i n d i c a t e t h e c l a s s of a l l mappings FN . The

c l a s s 7 of a l l mappings FN i s c a l l e d a n e a r - r i n g r e g u l a r i t y i f it

s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s :

(Al) I f a : N + N ' i s a near - r i n g epimorphism and n E N , t h e n

FN, ( W ) = 4 F N ( " ) ) .

(A2) I f I 4 N and i E I, t h e n F I ( i ) c F N ( i ) .

(13) I f n1 ,n2 E N and n2 E FN(nl), t h e n FN(nl+n2) < FN(nl) . The f o l l o w i n g n o t a t i o n s w i l l b e used throughout t h i s a r t i c l e :

1. I f T i s a near - r i n g r e g u l a r i t y , an element n E N i s F N - r e g u l a r

i f n E F N ( n ) .

2 . A near - r i n g N i s F- r e g u l a r i f every n E N i s F N - r e g u l a r .

3 . The c l a s s of a 1 1 F - r e g u l a r n e a r - r i n g s w i l l be deno ted by F.

4 . I 4 N i s F N - r e g u l a r i f i E F N ( i ) f o r a l l i E I.

5. I 4 N i s F - r e g u l a r i f i E F I ( i ) f o r a l l i E I, i . e I E IF.

For any n e a r - r i n g N we d e f i n e t h e s u b s e t U(N) = {n E N : <n> i s

F N - r e g u l a r ) , where t n > i s t h e i d e a l g e n e r a t e d by n i n t h e

n e a r - r i n g N . T h i s means t h a t M(N) c o n s i s t s of a l l e l emen t s of N

t h a t g e n e r a t e an F N - r e g u l a r i d e a l i n N . We can now prove t h e

f o l l o w i n g theorem.

Theorem 2.1. For any n e a r - r i n g N , U(N) i s t h e l a r g e s t F N - r e g u l a r

i d e a l i n N .

Proof. From t h e d e f i n i t i o n of U(N) it i s c l e a r t h a t U(N) i s

IN- r e g u l a r . To prove t h a t M(N) c o n t a i n s a l l t h e FN - r e g u l a r

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i d e a l s i n N , l e t x E I, where I i s an FN- regu la r i d e a l i n N.

S ince <x> g I it f o l l o w s t h a t a E FN(a) f o r every a E <x>.Thus

<x> i s F N - r e g u l a r and x E U(N). The theorem w i l l t h u s be proved

i f M(N) i s an i d e a l i n N . Le t m,n E U(N), i . e . <m> and <n> a r e

FN- r e g u l a r i d e a l s of N . Le t p E <m-n>. S ince <m-n> E <m> + a>,

t h e r e e x i s t s a E <m> and b E <n> such t h a t p = a+b. Because <m>

i s F N - r e g u l a r and p-b E < m > , we have t h a t p-b E FN(p-b) .

Let B : N + N/ = N / be t h e n a t u r a l epimorphism from N on to N ' .

As a consequence of (Al) we have t h a t B(FN(p-b)) = FN,(B(p-b)) =

FNl (B(p))=B(FN(p)) , which impl i e s t h a t p b E FN(p) + and,

there fore,^ E FN(p) + . Let p = pl+bl, where pl E FN(p) and

bl E . But p-pl= pl+bl-p1 E g <n> . Hence p-p1 E FN(p-pl)

and, s i n c e pi E FN (p) , we can deduce from (A3) t h a t

FN(p-pl) C FN(p) and t h u s t h a t p-p1 E FN(p) . S ince FN(p) i s an

a d d i t i v e group we have t h a t p E FH(p) , which proves t h a t

m-n E M(N) .To prove t h a t M(N) i s a normal subgroup, l e t m E M(N)

and n E N . S i n c e n + <m> = <m> + n , t h e r e e x i s t s a E <m> such

t h a t n+m = a+n. Thus <n+m-n> g <a> <m>. S ince <m> i s

FN- r e g l a r it proves t h a t n+m- n E H(N) .Next , l e t m E M(N) . Then

mn E <m> f o r every n E N and <mn> E <m>. S ince <m> i s

F N - r e g u l a r , it fo l lows t h a t mn E H(N) f o r a l l n E N . S i m i l a r l y ,

i f m E M(N) , t h e n n(n/+m)-nnj E <m> f o r every n and n / i n N , and

we have t h a t <n (n /+m)-nn l> 5 <m> . Consequently < n ( n # + n ) - n n l > is

FN- r e g u l a r .This completes t h e p roof .

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Corollarv 2.2. A near- ring N will be F- regular iff N =U(N) Thus

N 6 F iff N = U(N) .

3. ELEMENTVISE CHARACTERISATION OF A RADICAL CLASS

The method whereby a class of near-rings can be identified by a

certain property of its elements was developed by Xiegandt (cf.

[I31

If 7 is a certain property that an element n of a near-ring N

possesses, then n is said to be a 7- element. [We consider the

zero- element of N to be a ?- element.]

Notation

1. A near- ring N is called a P- near-ring if every element of N is

a 7- element.

2 . An ideal I 4 N is a P- ideal if every element of the ideal is a

7- element.

3. The class of all 7- near- rings is denoted by IRp.

Let N be a near-ring with I 4 N. Consider the following proper-

ties that 7 may satisfy:

(Bl) n E N a ?-element of N => n+I a ?-element of N/I.

(B2) n E I a 7-element of I => n a ?-element of N.

(B3) n E I and n a ?-element of N => n a ?-element of I.

(B4) n+I a P- element of N/I and I a ?- ideal =>n a ?-element of N.

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The following theorem was originally proved by Wiegandt [I31 f o r

associative rings and then extended by Gerber [5] t o be valid in

n-groups and thus also in near- rings.

Theorem 3.1. If P s a t i s f i e s properties (Bl) , (B2) and (B4), then

Rp i s a XI-radical c lass . I f , furthermore, P s a t i s f i e s property

(B3) then Rp i s a hereditary radical class.

In t h i s paper we w i l l consider an element of the near-ring t o

have the property P i f it i s FN- regular, i . e . n E N has the

property P i f n E FN(n).

In $ 2 we denoted the class of a l l F-regular rings by F. Thus a

near- ring N E ff i f every element of N has the property P, i . e .

every element of N i s FN-regular. It i s important t o note tha t

I E 5 where I a N i f every element of 1 i s PI-regular. We can now

proceed to prove the following theorem, implementing the tech-

nique described above.

Theorem 3.2. Let N be a near- ring and I a N . Then IF = {N : N a

F- regular near- ring) i s a HA- radical class.

Proof. We f i r s t prove (Bl) . Let a E N where N E ff , i. e.

a E FN(a). Let ll : N + N/I be the natural epimorphism from N

onto N/I. From (ill) we have: FNII(n(a)) = U(PN(a)).

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Thus FNII(a+I) = (FN(a) + I ) / 1 = {x+I; X E FN(a)}.

Since a E FN(a), it follows that a+I E FNII(a+I), which proves

that a+I i s a P-element of N/I.

Next we prove (B2). Let a E I where I E 5, i . e . a i s a P- element

of I. From (A2) we have that a E FI(a) < FN(a) , which proves

that a i s a P- element of N .

Lastly we prove (B4). Let a+I E N/I where N/I E F and I E F.If a

E I the proof i s completed.Suppose then a j! I.Let tl : N + N/I be

the natural epimorphism such that II(a) = a+I. From (Al) we have:

FN,I(II(a)) = II(FN(a)). Thus a t 1 E F ( 1 = ( F a ) + I ) / I . N/I

This implies that a E FN(a) + I. Let a = x+I where x E FN(a)

Since I i s a P- ideal we have that a-x E FN(a-x). It then follows

from (A3) that FN(a-x) FN(a). Subsequently a-X E FN(a) and

thus a E FN(a) , which concludes the proof.

The following lemma proves useful in the discussion on certain

specific regular i t ies to follow.

Lemma 3.3 . If y : N -I N 1 i s a near-ring epimorphism, then

~ ( < x > ~ ) = < y ( ~ ) > ~ , where < x > ~ and < V ( X ) > ~ , are ideals generated

i n N and N / respectively.

Proof. Let x E N . Thus f(x) E f ( < x > ) , which implies tha t

<P(x)> c P ( W .

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Conversely, l e t y E p(<x>), say y = p(a) where a E <x>.Let B 4 N'

w i t h p(x) E 8 , tha t i s x E f i (B) Q N . Thus a E tx> C f l ( B ) and

subsequently y = p(a) E p(<x>) C p ( i l ( B ) ) = B fo r a l l B Q N f

with p(x) E B . This proves tha t y E n{B Q N) : p(x) E B) and

consequently y E <p(x)>, which proves the lemma.

The question may a r i s e as t o which conditions on the regulari ty 3

are necessary fo r the F-radical F(N) and the ideal M(N) t o coin-

cide i n every near-ring N . The following theorem addresses t h i s

problem . Theorem 3.4. Let T be any regulari ty. The following three

statements are equivalent:

( i ) F(N) = M(N) fo r any near-ring N.

( i i ) M(M(N)) = U(N) fo r any near-ring N .

( i i i ) F(N) = 0 implies M(N) = 0 for any near-ring N .

Proof. Since 5 = {N : M(N) = N) i s a Kurosh- Amitsur radical

c lass , we have F(F(N)) = F(N) fo r any near- ring N . Hence, i f

F(N) = M(N), then M(M(N)) = M(N). Suppose now M(M(N)) = M(N) fo r

any near-ring N . This implies M(N) E F and hence U(N) g F(N).

Since F(N) g M(N) fo r every N , we have the equivalence of ( i ) and

( i i ) . We only have to show ( i i i ) => ( i ) since ( i ) => ( i i i ) i s

c lear . Since F (N/F(N)) = 0, it follows from ( i i i ) tha t

M(N/B(N)) = 0. To complete the proof, we show that i f N i s any

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near-r ing, then U(N) i s the smallest ideal of N such tha t

U(N/U(N)) = 0. Suppose A q N and U(NJA) = 0. Let r : N -I NJA be

the canonical epimorphism. and l e t x E M(N) be arbi t rary . Now

x E FN(x) and, since T i s a regulari ty, we have

FNIA(n(x)) = r ( F N ( x ) ) Hence r(x) E F ( r (x)) and, therefore, NIB r(M(N)) i s an F -regular ideal . Since U(N/A) i s the greatest

N / A FNJA- regular ideal , we have r(M(N)) E Y(N/A) = 0. From t h i s we

have U(N) A and consequently, U(N) E F(N).

4 . EXAMPLES

We w i l l now proceed t o show tha t some well-known regu la r i t i e s

indeed sa t i s fy the conditions fo r a near- ring regulari ty.

1) Ouasiregular (cf . [7] p. 89 and [8] p.48). For a subset S of

N , <S>, (<S)>e) denotes the ideal ( l e f t - ideal) generated by the

se t S. The element z E N i s called quasiregular if

z E <N-N~>~,where <N-Nz>e i s the l e f t ideal generated by the se t

{n - nz : n E N ). Let FN(x) = <N-Nx>!. It follows eas i ly tha t

conditions (Al) and (A2) are sa t i s f i ed f o r quasiregularity. To

prove (A3), l e t nl,n2 E N such tha t n2 E FN(nl) = <N-Nnl>e. Let

n E N , then

n(n1+n2) - n = nnl+(-nnl + [n(n1+n2) - nnl] + nnl) - n

= nnl + y - n ; where y E <n2>e.

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Hence <{n(nl+n2) - n : n E N)>! = <{nnl + y - n : n E N)>!

= <innl-n+y' : n E N)>! ; y' E <n2>t

Since <nZ>l E <Nnl-N>! , it implies tha t

<N(n1+n2) - N > e E <Nnl-N>!, i . e . < N - N(nl+n2)>t C <N-Nnl>!,which

proves (A3). If N i s any zero-symmetric near-ring it follows from

[7] , Theorem 5.37(c) tha t U(N) = Jo(N) , where Jo(N) denotes the

Jo-radical of the near-ring N . Let 9 = {N : N = Jo(N)) = {N : N

i s quasiregular). From Theorem 3.2 we have tha t J' i s a radical

class with radical J(N) f o r any near-ring N . In [6] Kaarli gave

an example of a zero-symmetric near-ring N such tha t

Jo(N) # Jo(Jo(N)) . It follows from Theorem 3.4 tha t in general

JO(N) # m.

2) G Begularitv. Ramakotaiah [9] introduced the concept of

G- regulari ty and used it t o define the Brown-McCoy radical in

near- rings . An element z E N i s called G-regular if z E tN-Nz>

The proof tha t G- regulari ty i s a near- ring regulari ty proceeds on

l ines similar t o (1) above. As in [8] we c a l l an ideal I a N

G- regular i f each element of I i s G-regular. In [9] Ramakotaiah

defined the largest G-regular ideal of the near-ring N and

denoted it by B(N) From Theorem 3.4 we have tha t B = {N : N =

B(N)) = {N : N i s G-regular) i s a radical class with radical B(N)

fo r any near- ring N . It follows from Kaarli [6] tha t there ex-

i s t s a near- ring N such tha t B(N) # B(B(N)) and from Theorem 3.4

it follows tha t i n general B(N) # B(N).

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3) Leftpreerularitv. In [3] Divinsky introduced the concept of

P r e g u l a r i t y fo r associative rings and in [7] Bhandari and Saxena

extended t h i s concept to near-rings. Let FN(x) = < N X > ~ . We show

that t h i s i s a near-ring regulari ty. Suppose tha t a : N 4 N ' i s

a near-ring epimorphism .The val id i ty of condition (Al) follows

eas i ly from Lemma 3.3. Next, l e t I Q N and i E I. Thus FN(i) =

< N b t 2 <Ii>t = "( i ) , which proves (A2). To prove (A3), l e t

n1,n2 E N such tha t n2 E FN(nl) = <Nnl>t.

Thus FN(n1+n2) = <N(n1+n2)>t. Let n E N.

Then n(nl+n2) = nn 1 + -nnl + [n(n1+n2) - nnl] + nnl = nnl + y ;

y E <n2>e. Thus <N(nl+n2)>& <Nnl>e + <Y>e C <Nnl>L + <n2>! 5

cNnl>t since n2 E <Nnl>e. This proves tha t FN(n1+n2) 5 FN(nl)

and consequently tha t l e f t Y- regulari ty i s a near- ring

regulari ty.

As a d i r ec t consequence of Theorem 3.2 we have the following

resul t of [ll] ,

Theorem 4.1 ( [I] , Theorem 3.2). The class of a l l l e f t 3- regular

near- r ings i s a KA- radical class.

4) Pseudo-reeularitv. In [4] Divinsky introduced the concept of

pseudo-regularity f o r associative rings and in [I21 Bhandari

extended t h i s t o near- rings. If N i s any near- ring and x E N , we 2 define FN(x) = <Nx - Nx >. An element z E N i s cal led pseudo

2 - regular i f z E <Nz - Nz >. We show that pseudo- regulari ty i s a

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near- ring regularity. The proofs for (Al) and (A2) are simple.

To prove (A3), let n1,n2 E N such that n2 E PN(nl) = <Nnl - 2 Nnl>.Let n E N. Then n(nl+n2) - nnlE <n2>, say n(nl+n2) - nnl

2 Thus n(nl+n2) = (x+nn ) n +n 1 ( 1 2)

= y + nnl , . where y E <y2>.

2 2 Hence <n(n1+n2)-n(n1+n2) ,n E N > = <z+nnl-nnl,n E N> 2 2 5 t z > + <nnl - nnl, n E N> C <Nnl - Nnl).

This proves that FN(nl+n2) g FN(nl) and that pseudo-regularity is

a near- ring regularity.

5 ) Behrens- reenlarity. In [I] Behrens introduced a radical class

B for assosiative rings that lies properly between the Jacobson

radical class 3 and the Brown-HcCoy radical class E. Let 2

PN(x) = <X -x> for x E N. The proofs for (dl) and (A2) are ob-

vious. 2 To prove (A3), let n1,n2 E N such that n2E FN(nl)' <nl - nl>,

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A GENERALISATION OF REGULARITIES IN NEAR-RINGS

2 2 The re fo re ~ ( n ~ + n ~ ) ~ - (n 1 +n 2) > g t n l - nl> + t x > < t n l - nl>

Thus FN (n1+n2) FN(nl) , which proves (A3) . 2 I f we d e f i n e x E N t o be Behrens-regular when x E t x - x > , i t t h e n

f o l l o w s from Theorem 2 . 1 t h a t U(N) i s t h e l a r g e s t Behrens- r e g u l a r

i d e a l i n N . For any n e a r - r i n g N , we c a l l t h i s l a r g e s t

Behrens- r e g u l a r i d e a l i n N t h e Behrens- r a d i c a l and deno te it by

X(N). From Theorem 3.2 we have t h a t t h e c l a s s {N : N = X(N)) i s

a KA-radical c l a s s . S ince f o r any n e a r - r i n g N and z E N , we 2 always have t z - z > C <N-Nz>, it fo l lows t h a t X(N) < B(N) . This

containment can be p r o p e r , a s is ev iden t from t h e f o l l o w i n g

example.

m. Consider S3 = {O,a ,b , c ,d , e ) wi th a d d i t i o n and mul t i -

p l i c a t i o n o p e r a t i o n s on S3 de f ined a s :

+ O a b c d e O a b c d g

O O a b c d e 0 0 0 0 0 0 0

a a O e d c b a 0 0 0 0 0 0

b b d O e a c b O O O O O O

c c e d O b a ~ 0 0 0 0 0 0

d d b c a e O d O a c b e d

e e c a b O d e O a b c d e

Th i s i s a n e a r - r i n g having no proper i d e a l s [2]. It i s e a s y t o

show t h a t B(N) = N and K(N) = 0. Hence K(N) # B(N).

UPERENCES rll Behrens, A.E. N i c h t a s s o z i a l t i ve Binge, Math. Annalen

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[2] Clay, J .R. Near-rings on groups of low order. l a t h . Z. 104(1968), 364- 371.

[3] Divinsky, N . J . P-regularity. Proc. Amer. l a t h . Soc. 9 (1958) , 62- 71.

[4] Divinsky, N . J . Pseudo-regularity. Canad. J . l a t h . 7(1955) , 401- 410

[5] Gerber , G . K . Radicals of R-groups defined by means of e 1 ements. Near- r i n g s and near- f i e l d s (North Holland 1987) 87- 96

[6] K a a r l i , K . Survey on the radical theory of near-rings. Proc . of Conf . on Rad ica l Theory (Krems 1985)

[7] P i l z , G . Near-rings. North-Holland, 1977.

[8] Ramakotaiah D . , Radicals for near-rings. l a t h . Z. 97(1967) , 45- 56.

[9] Ramakotaiah D . A radical for near-rings. Archiv. d e r l a t h . 23 (1972) , 482- 483.

[ lo] Roos, C . lings and Regularities. Ph.D d i s s e r t a t i o n D e l f t Holland (1975)

[ll] Saxena, P.K. and Bhandari , I . C . P-regularity for near-rings. I n d i a n J . pure and app l . Math. 12(8) ( l 9 8 l ) , 938- 944.

[12] Saxena, P.K. and Bhandari , N.C. Pseudo- regularity for near-rings. I n d i a n J . pu re and a p p l . l a t h . , l 3 ( l 9 8 2 ) , 1409- 1412

[13] Wiegandt, R . Radicals of rings defined by means of elements Si t zungsbe r d e r O s t e r r e i c h Bkad. d e r Wiss., Mathem. naturw. Klas se , Vo1.184, 975

Received: December 1987 Revised: August 1988

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A generalisation of regularities in near-rings