Content uploaded by Nimmagadda Venkata Nagendram
Author content
All content in this area was uploaded by Nimmagadda Venkata Nagendram on Jan 22, 2015
Content may be subject to copyright.
Int. J. of Contemporary Mathematics
Vol. 2, No. 1, June 2011
Copyright
Mind Reader Publications
ISSN No: 0973-6298
On
IFP IDEALS IN NOETHERIAN REGULAR δ-NEAR-RINGS
N V Nagendram
1
Lakireddy Balireddy College of Engineering, L B Reddy Nagar
Mylavaram, Krishna District, AP., India
Email: nvn220463@yahoo.co.in
T V Pradeep Kumar
2
ANU college of Engineering and Technology,
Acharya Nagarjuna University,Nagarjuna Nagar, AP., India
Email: pradeeptv5@gmail.com
Dr. Yanumula Venkateswara Reddy
3
ANU College of Engineering and Technology,
Acharya Nagarjuna University,Nagarjuna Nagar, AP., India
Abstract
A Noetherian regular δ-near-ring N is called an IFP Noetherian regular δ-
near-ring provided that for all a, b, n Є N , ab = 0 implies anb = 0. In this
observation , the IFP (Insertion Factors Property) condition in a Noetherian
regular δ-near-ring is extended to the ideals in Noetherian regular δ-near-ring
as well as in Near-Rings also. If N / P is an IFP Noetherian regular δ-near-
ring, where P is an ideal of a Noetherian regular δ-near-ring N, then we call P
as the IFP – ideal of N. In this paper we obtained relations between prime
ideals and IFP ideals are investigated in Noetherian regular δ-near-rings.
Keywords :
Near-ring, Prime ideal, IFP, Nilpotent element, Equi prime near-ring, Noetherian near-
ring, regular near-ring, Noetherian Regular δ-near-ring.
Subject Classification
: 2000 AMS 16 Y 30.
Section 1. Introduction
IFP near-rings have been studied by several authors since they were introduced in
[ 3 ] and [ 11 ]. In this study, the IFP property in Noetherian regular δ-near-rings is
extended to the ideals of near-rings and these ideals, say IFP-ideals, of some certain
classes of Noetherian regular δ-near-ring in near-rings are considered. It shown that a
On IFP IDEALS IN NOETHERIAN REGULAR δ-NEAR-RINGS
54
right permutable or left permutable equi prime Noetherian regular δ-near-ring which has
no non-zero nilpotent elements, it was showed that if N is a finite Noetherian regular
δ-near-ring , then N is a near-field if and only if N is an equi prime Noetherian regular
δ-near-ring, and has no non-zero nilpotent elements. Using this result, we obtained “ If N
is a right permutable or left permutable finite Noetherian regular δ-near-ring, then N is a
near-field if and only if N is an equi prime Noetherian regular δ-near-ring”. Besides, it is
proved that every completely semi prime ideal of a zero-symmetric Noetherian regular
δ-near-ring is an IFP-ideal and using this result some conclusions are obtained
concerning under what conditions 0-prime and 3-semi prime ideals are IFP-ideals and
the concept of IFP ideals occurs naturally in some Near-Rings , such as p-near-rings,
Boolean near-rings, weekly (Right/Left) permutable near-rings, left(right) self
distributive near-rings, left (right) strongly regular near-rings and left(w-) weakly regular
near-rings. Finally, some certain Noetherian regular δ-near-ring classes that bear the
property of IFP-ideal on itself naturally are given.
In this section we give some definitions, examples, notations which are useful in later
sections.
Definition : 1.1 A Near – Ring is a set N together with two binary operations “+” and “.”
Such that (i) (N, +) is a group not necessarily abelian (ii) (N, .) is a semi group and (iii)
٧ n
1
, n
2
, n
3
belongs to N ,(n
1+
n
2
). n
3
=
(n
1 .
n
3
+ n
2 .
n
3
) i.e., right distributive law.
Definition : 1.2 A Commutative ring N with identity is a Noetherian Regular δ-Near Ring
if it is Semi Prime in which every non-unit is a zero divisor and the Zero ideal is a Product
of a finite number of principle ideals generated by semi prime elements and N is left
simple which has N
0
= N, N
e
= N.
Definition :1.3 A triple (S, +, .) is called a semi-Noetherian Regular δ-Near Ring if (S, +)
and (S, .) are Semi-groups and is distributive over +. And also an element 0ЄS is called a
zero of the semi-Noetherian Regular δ-Near Ring (S, +, .) if x+0 = 0+x=x , x.0 = 0.x = 0 ,
x ЄS.
Definition :1.4 N belongs to
ŋ
is called “Zero-symmetric (const.) if N = N
0
( N = N
C
)
respectively.
ŋ
0 ,
ŋ
C
stands for the classes of all Zero-symmetric (constant) Near – Rings.
Definition : 1.5 Element n € N is said to be nilpotent if there is a positive integer m such that
n
m
= 0 where n
m
stands for n.n.n.n….(m factors). 0 is called trivial nilpotent element.
Definition : 1.6 If N* ={ N \ (0), . } is a group then N is called Near–Field & denoted by
“nf ”.
Definition : 1.7 Equi Prime Near-Ring : N is said to be Equi Prime if its zero ideal is equi
prime, that is, if there exists an element 0 ≠ a Є N, such that, for n
1
, n
2
Є N, ann
1
= ann
2
,for
ann
1
Є N, implies n
1
= n
2
.
Or
we can define as
P
is completely semi prime to
N is called equi prime ideal if a Є N - P
N V Nagendram, T V Pradeep Kumar and Yanamula Venkateswara Reddy
55
and x, y Є N such that anx – any Є P for all n Є N then x – y Є P.
Definition: 1.8 IFP –Insertion of Factors Property : N is said to fulfill the insertion-of-
factors property (IFP) provided that for all a, b, n Є N such that ab = 0 implies anb = 0.
Such near-rings are called IFP near-rings.
Definition : 1.9 If
P
is completely semi prime to N and N/P is an IFP near-ring, then the
ideal P is called an IFP-ideal of a Noetherian Regular δ-Near Ring N.
Definition : 1.10 The ideal P of a Noetherian Regular δ-Near Ring N is called a 0-
Prime ideal if for every A, B are completely semi prime to N, AB is subset of P implies
either A is subset of P or B is subset of P.
Definition : 1.11 If P is completely semi prime to N is called 3-semiprime or 3-prime
ideal if for every a, b Є N aNb is subset of P or aNa is subset of P implies a Є P or b Є P
(a Є P).
Definition : 1.12 If for all a, b Є N , ab Є P so a
2
ЄP implies a Є P or b Є P (a Є P) then
P is completely semi prime to N is called a completely prime (completely semi Prime)
Definition : 1.13 If for all a, b, c, d Є N, abc = acb (respectively, we have abc = bac ,
abcd = acbd) then N is called a right Permutable (or left Permutable) in a Noetherian
Regular δ-Near Ring N.
Section 2:
Some Fundamental Results on IFP Ideals in a Noetherian Regular- δ-
Near Rings :
Lemma : 2.1 [ 3.1 of [ 1] ]:Let N be a zero-symmetric finite Noetherian Regular δ-Near
Ring. Then N is a Near-Field if and only if N is Equiprime and has no non-zero nilpotent
elements.
Proposition.: 2.2 If N is a right (or left) permutable 3-prime Noetherian Regular δ-near-
ring, then N has no non-zero nilpotent elements.
Proof. Let ab = 0 for a, b Є N. If N is right permutable (resp. left permutable), then
abN = aNb = 0 (resp. Nab = aNb = 0). Since N is 3-prime, then a = 0 or b = 0, i.e. N
is a completely prime Noetherian Regular δ-near-ring. Hence N has no non-zero nilpotent
elements.
Corollary : 2. 3 Let N be a finite near-ring.
a) If N is a zero symmetric right permutable near-ring, then N is a near-field if and only
if N is an equi-prime near-ring.
On IFP IDEALS IN NOETHERIAN REGULAR δ-NEAR-RINGS
56
b) If N is a left permutable near-ring, then N is a near-field if and only if N is an
equi-prime near-ring.
Proof. If N is left permutable, then for all n Є N , n0 = n00 = 0n0 = 0, i.e. N is
Zero-symmetric. Hence the result follows from Lemma 3.1 of [1].
Proposition : 2.4 If P is an IFP-ideal and a 3-(semi) prime ideal of N, then P is a
completely (semi) prime ideal.
Proof. Let ab Є P for a, b Є N. Since P is an IFP-ideal, then aNb is subset of P. Hence a
Є P or b Є P, since P is a 3-prime ideal. Therefore P is a completely prime ideal. To
prove the semi prime case, it is enough to take a = b. Hence Proved the Proposition.
Theorem : 2.5 Let P be a completely semi prime ideal of N. Then P is an IFP - ideal.
If P is an IFP-ideal of a Noetherian Regular δ-near-ring N, then (P : P) read as P over P is
an ideal of N. Furthermore (P : P) is also an IFP-ideal.
Proof. Assume P is a completely semi prime ideal of N and ab Є P for a, b Є N. It is
easily seen that NP is subset of P, since N is zero-symmetric. Then (ba) Є = baba ЄNPN
is subset of P and then ba Є P since P is completely semiprime. Hence (anb)
2
= anbanb Є
NPN subset of P, for all n Є N, whence anb Є P since P is completely semi prime ideal of
a Noetherian Regular δ-near-ring N. Therefore P is an IFP-ideal.
To prove (P : P) is subset of N, it is enough to show that (P : P)N subset of (P : P). Let y
Є (P : P)N. Then there exist an a Є (P : P) and an n Є N such that y = an. Since a Є (P :
P), then ap Є P for all p Є P. Since P is an IFP-ideal, then anp Є P for all n Є N. Then yp
Є P for all p Є P. Hence y Є (P : P). Now, we show that (P : P) is an IFP-ideal. Assume
xy Є (P : P) for x, y Є N. Then xyp Є P for all p Є P. Since P is an IFP-ideal, xnyp Є P
for all n Є N and for all p Є P. Therefore, xny Є (P : P), which completes the proof of
Theorem.
Note : 2.6 Let N be a medial Noetherian Regular δ-near-ring and P a 3-prime ideal of N.
Then P is an IFP-ideal.
Note : 2.8 Let N xny Є Ro \ R1 be a reduced left (w-) weakly regular near-ring and
let P subset of a Noetherian Regular δ-near-ring N. Then P is an IFP-ideal of N.
Lemma : 2.9 Let P subset of a Noetherian Regular δ-near-ring N. Then,
a) If N is right permutable, then P is an IFP-ideal. b) If N is left permutable, then P is an
IFP-ideal.
c) If N is right self distributive, then P is an IFP-ideal. d) If N Є Ro is left self
distributive, then P is an IFP-ideal.
N V Nagendram, T V Pradeep Kumar and Yanamula Venkateswara Reddy
57
Proof. For a, b Є N, assume ab Є P. Then for all n Є N; a) anb = abn Є PN is subset of P.
b) anb = nab Є NP is subset of P, since N Є R
o
. c) anb = abnb Є PN is subset of P. d) anb
= anab Є NP is subset of P, since N Є R
o
. Hence Proved Lemma.
Theorem : 2.10 Let N be a medial near-ring and P is subset of N. Then, a) If N is
regular, then P is an IFP-ideal. b) If N is right strongly regular, then P is an IFP-ideal. c)
If NЄRo is left strongly regular, then P is an IFP-ideal. IFP Ideals in Near-Rings
Proof. For x, y Є N, assume x.y Є P.
a) Since N is Noetherian Regular δ-Near Ring which is regular, then there exist a, b Є N
such that x = xax and y = yby. Then for all n Є N, xny = xaxnyby = x(axn)y(by). Since N
is medial, then xny =xy(axn)by Є PN is subset of P.
b) Since N is right strongly regular, then there exist a, b Є N such that x = x
2
a and
y = y
2
b. Then for all n Є N, xny = xxanyyb = x(xan)y(yb). Since N is medial Noetherian
Regular δ-Near Ring , then xny = xy(xan)(yb) Є PN is subset of P.
c) Since N Є Ro is left strongly regular, then there exist a, bЄN such that x = ax
2
and y = by
2
. Then for all n Є N, xny = axxnbyy = (ax)(xnb)y(y). Since N is medial
Noetherian Regular δ-Near Ring , then xny = (ax)y(xnb)y = a(xy)xnby Є NPN is subset
of P.
References
[ 1 ]. Akın Osman Atag¨un_ “IFP Ideals in Near Rings” , Hacettepe Journal of
Mathematics and Statistics Volume 39 (1) (2010),17– 21.
[ 2 ] Argac, N. and Groenewald, N. J. Weakly and strongly regular near-rings, Alg.
Coll. 12, 121–130, 2005.
[ 3 ] Bell, H.E. Near-rings in which each element is a power of itself, Bull. Austral.
Math. Soc. 2, 363–368, 1970.
[ 4 ] Birkenmeier, G. and Groenewald, N. J. Near-rings in which each prime factor is
simple, Math. Pannon. 10, 257–269, 1999.
[ 5 ] Dheena, P. and Rejeswari, C. Weakly regular near-rings, Indian J. Pure and Appl.
Math. 28, 1207–1213, 1997.
[ 6 ] N V Nagendram,Dr T V Pradeep Kumar and Dr Y V Reddy on “Noetherian
Regular Delta Near Rings and their extensions”,International Journal of Contemporary
Mathematical Sciences,Bulgaria,IJCMS5-8-2011,Vol.6,2011,No.6,255-262.
[ 7 ] N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy on “Strongly Semi
Prime Noetherian Regular Delta Near Rings(SSPNR-delta-NR)”, accepted and to be
On IFP IDEALS IN NOETHERIAN REGULAR δ-NEAR-RINGS
58
published in the month of June 2011 by International Journal of Contemporary
Mathematics (IJCM) thru’ Mind Reader Publications(MRP), New Delhi,INDIA.
[ 8 ] N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy on “Semi Noetherian
Regular Matrix Delta Near Rings and their extensions(SNRM-delta-NR)”,
International Journal of Advances in Algebra(IJAA),JORDAN accepted and to be
published in the month of June 2011 by AA. Thru’ Research India Publications(RIP),
Rohini, Delhi-110 015 INDIA.
[ 9 ] N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy on “A Note on
Bounded Matrices over a Noetherian Regular delta Near Rings"(BMNR-δ-NR)”,
accepted and to be published in the month of June 2011 by International Journal of
Contemporary Mathematics (IJCM) thru’ Mind Reader Publications(MRP), New
Delhi,INDIA.
[ 10 ] Pilz, G. Near-rings (North-Holland, 1983).
[ 11 ] Veldsman, S. On equiprime near-rings, Comm. Algebra 20, 2569–2587, 1992.
[ 12 ]Y.V.Reddy And C.V.L.N Murthy ‘On Strongly Regular Near-Rings’ Proc.
Edinburgh Math.Soc.27(1984),62-64.
