International Journal of Mathematical Archive6(4), 2015,
3138
Available online through www.ijma.info
ISSN 2229 – 5046
International Journal of Mathematical Archive 6(4), April – 2015 31
A NOTE ON DIVIDED NEARFIELD SPACES AND PSEUDO – VALUATION NEARFIELD SPACES OVER REGULAR NEARRINGS (DNF
PVNFSO
NR) Dr. N V NAGENDRAM*
Professor in Mathematics. Department of Mathematics(S&H), Kakinada Institute of Technology & Science Tirupati (V), Divili, Peddapuram (Mandal), East Godavari District Pin: 533 433, Andhra Pradesh. India.
(Received On: 180215; Revised & Accepted On: 230415)
ABSTRACT
L
et N be a commutative nearfield space with 1 and T(N) be the total quotient nearfield space such that Nil(N) is a divided prime ideal of a nearfield space N. Then N is called a
φ
chained nearfield space (
φ
CNF) if for every x, y
∈
N \ Nil(N) either x

y or y

x. Also, N is called a
φ
pseudo –valuation nearfield space (
φ
PVNF) if for every x, y
∈
N \ Nil (N) either x

y or y

xm where for each nonunit element m
∈
N. We show that a nearfield space N is a
φ
PVNF iff Nil (N) is a divided prime ideal and N/Nil(N) is a pseudovaluation domain. Also, we show that every over nearfield space of a Quasilocal nearfield space N with maximal ideal M is a
φ
PVNF iff N (v) for each v
∈
(M: M) \ N iff every over nearfield space of N is a quasilocal iff every
φ
CNF between N and T(N) other than N and (M : M) is of the form N
p
for some nonmaximal prime ideal P of N. Among other results, we show that if A is an overnearfield space of a
φ
PVNF and J is a proper ideal of A, then there is a
φ
CNFC between A and T(N) such that JC
≠
C. Also, we show that the integral closure N
c
of a nearfield space N in T(N) is the intersection of all the
φ
CNFs between N and T(N).
Subject Classification Code:
MSC (2010):
16D25, 54G05, 54C40.
Keywords:
Nearfield, Near–field space, total quotient nearfield space, divided prime ideal, quasi nearfield space, Quasilocal nearfield space, overnearfield space, pseudo –valuation nearfield space, maximal ideal, maximal ideal.
SECTION 1: INTRODUCTION
Throughout this paper, N denotes as Nearfield space has zero symmetric nearring with identity. We begin by recalling some background material. With reference to ([1], [4]) the author generalized the study of pseudovaluation domains to the context of extending to arbitrary nearfield spaces possibly with nonzero zero divisors. For a nearfield space N with total quotient nearfield space T(N) such that Nil(N) is divided prime ideal of N, we define a map
φ
: T(N)
→
K := N
Nil(N)
such that
φ
(a/b) = a/b
∀
a
∈
N and b
∈
n \ Z(N). Then
φ
is a nearfield homomorphism from T(N) into K, and
φ
is restricted to nearfield space N is also a nearfield homomorphism from N into K given by
φ
(x) = x/1
∀
x
∈
N. For an equivalence characterization of a
φ
PVNFS,
∀
n
≥
0
∃
a
φ
CNFS of krull dimension n that is not a PVNFS. In this paper, we show that a quasilocal nearfield space N with maximal ideal M is a
φ
PVNFS if and only if N(v) is a quasilocal nearfield space for each v
∈
(
M
:
M
) \ N if and only if every overnearfield space of N is quasilocal nearfield space if and only if every overnearfield space contained in (
M
:
M
) is quasilocal nearfield space if and only if each
φ
CNFS between N and T(N) other than (M : M) is of the form N
q
for some nonmaximal prime ideal P of N. Among the other results, we show that if A is an overnearfield space of a
φ
PVNFS and J is a proper ideal of A, then there is a
φ
CNFS C between A and T(N) such that
JA
≠
A
. Also show that the integral closure of nearfield space N in T(N) is the intersection of all the
φ
CNFS’s between N and T(N). The following notations will be used throughout. Let N be a nearfield space. Then T(N) denote the total quotient nearfield space of a nearfield space N.
Nil
(N) denotes the nearfield spaces of all nilpotent elements of N, and Z(N) denotes the set of zero divisors of N. If
J
is an ideal of N, then
Rad
(
J
) denotes the radical ideal of
J
in N.
Corresponding Author: Dr. N V Nagendram* Professor in Mathematics. Department of Mathematics(S&H), Kakinada Institute of Technology & Science Tirupati (V), Divili, Peddapuram (Mandal), East Godavari District Pin: 533 433, Andhra Pradesh. India.
N V Nagendram* / A Note On Divided NearField Spaces and Pseudo – Valuation NearField Spaces Over Regular NearRings (Dnf
PvnfsO
Nr) / IJMA 6(4), April2015. © 2015, IJMA. All Rights Reserved 32
I summarize some basic properties of PVNFSs and
φ
PVNFSs as below:
Property 1.1:
A PVNFS is a divided nearfield space and hence is quasilocal nearfield space.
Property 1.2:
A
φ
PVNFS is a divided nearfield space and hence is quasilocal nearfield space.
Property 1.3:
A sub–nearfield space is a PVNFS
iff
it is a
φ
PVNFS
iff
it is a PVD.
Property 1.4:
A nearfield space N is a PVNFS
if and only if
∀
a, b
∈
N, either a/b
∈
N or b/a
∈
N for each nonunit c
∈
N.
Property 1.5:
A nearfield space N is a
φ
PVNFS
if and only if
N
il
(N) is a divided prime ideal of N and
∀
a, b
∈
N \ N
il
(N), either a/b
∈
N or b/a
∈
N
∀
nonunit c
∈
N.
Property 1.6:
If N is a PVNFS or a
φ
PVNFS, then N
il
(N) and Z(N) are divided prime ideals of a nearfield space N.
SECTION 2: PRELIMINARY RESULTS AND EXAMPLES Definition 2.1:
A nearfield space N, with quotient nearfield space K of N is called a pseudovaluation domain [PVD] nearfield space in case each prime ideal P of N is strongly prime in the sense that xy
∈
P
∀
x
∈
K, y
∈
K
⇒
either x
∈
P or y
∈
P.
Definition 2.3:
A prime ideal P of a nearfield space N is said to be strongly prime in N if aP and bP are comparable under inclusion of nearfield spaces
∀
a, b
∈
N.
Definition 2.4:
A nearfield space N is called a pseudovaluation nearfield space (PVNFS) if each prime ideal of N is strongly prime. A PVNFS is necessarily quasilocal nearfield space.
Note 2.5:
A nearfield space is a pseudovaluation nearfield space (PVNFS) if and only if it is pseudovaluation domain [PVD].
Definition 2.6:
A prime ideal P of a nearfield space N is called divided if it is comparable under inclusion to every ideal of nearfield space N.
Definition 2.7:
A nearfield space N is called a divided nearfield space if every prime ideal of a nearfield space N is divided.
Definition 2.8:
A prime ideal Q of
φ
(N) is called a Kstrongly prime ideal if xy
∈
Q,
∀
x
∈
K, y
∈
K
⇒
either z
∈
Q or y
∈
Q.
Definition 2.9:
If each prime ideal of
φ
(N) is Kstrongly prime, then
φ
(N) is called a Kpseudovaluation nearfield space (KPVNFS).
Definition 2.10:
A prime ideal P of nearfield space N is called a
φ
strongly prime ideal if
φ
(P) is a Kstrongly prime ideal of
φ
(N).
Definition 2.11:
A prime ideal P of N is called a
φ
strongly prime ideal if
φ
(P) is a Kstrongly prime ideal of
φ
(N). If each prime ideal of nearfiled space N is
φ
strongly prime, then N is called a
φ
 pseudovaluation nearfield space (
φ
PVNFS).
Definition 2.12:
a nearfield space N is called a
φ
chained nearfield space (
φ
CNFS) if Nil(N) is a divided prime ideal of N and
∀
x
∈
N
Nil(N)
\
φ
(N) , we have x
1
∈
φ
(N).
Note 2.13:
A chained nearfield space (
φ
CNFS) is a division nearfield space and hence is quasilocal nearfield space. Hence,
∀
n
≥
0
∃
a
φ
CNFS of krull dimension n that is not a chained nearfield space.
Definition 2.14:
A proper ideal of a nearfield space N is called a divided ideal if J is comparable under inclusion to every principal ideal of N; equivalently, if J is comparable to every ideal of N. If every prime ideal of N is divided, then N is called a divided nearfield space.
Definition 2.15:
A prime ideal Q of a nearfield space B is branched if Rad (J) = Q for some primary ideal J
≠
Q of B.
N V Nagendram* / A Note On Divided NearField Spaces and Pseudo – Valuation NearField Spaces Over Regular NearRings (Dnf
PvnfsO
Nr) / IJMA 6(4), April2015. © 2015, IJMA. All Rights Reserved 33
Note 2.16:
A prime ideal Q of a nearfield space domain D is branched
iff
Rad (J) = Q for some ideal J
≠
Q of D. In the following result I will show that this result is still valid for divided nearfield spaces.
Definition 2.17:
An ideal of a nearfield space N is called regular if it contains a nonzero divisor of N. If every regular ideal of N is generated by its set of nonzero divisors, then N is called as Nagendram nearfield space.
Definition 2.18:
A nearfield space N has few zerodivisors if Z(N) is a finite union of prime ideals.
SECTION 3: RESULTS ON DIVIDED NEARFIELD SPACES AND PVNFS
In view of the proof of [5, Proposition 2.1], we see that the result in [5, proposition 2.1] valid iff assume that the nearfield space N is a divided nearfields domain. Hence, I state the following result without proof.
Proposition 3.1:
[5, Proposition 2.1] Let D be division nearfield space domain with maximal ideal K and krull dimension n, say K = Q
n
⊃
Q
n1
⊃
Q
n2
⊃
………
⊃
Q
1
⊃
{0}, where the Q
j
s are the distinct prime ideals of division nearfield space domain D. Let, j, m, d
≥
1 such that 1
≤
j
≤
m
≤
n. Choose z
∈
D such that Rad ((y)) = Q
j
. Let P:= Q
m
and
I := y
j+1
D
P
. Then (i)
I is an ideal of nearfield space domain D and rad (I) = Q
j
. (ii)
N:= D/I is a divided nearfield space with maximal ideal K/I, Z(N) = Q
m
/I, and Nil (N) = Q
j
/I. Furthermore, v:= y + I
∈
Nil (N) and v
d
≠
0 in N. (iii)
Dim (N) = n – j. (iv) if j
≤
m
≤
n, then Nil (N) is properly contained in Between Z(N) and M/I.
Proposition 3.2:
Let N be a divided nearfield space and let Q be a prime ideal of N such that Q
≠
Nil (N). Then Q is branched
if and only if
Rad (J) = Q or some ideal J
≠
Q of nearfield space N.
Proof:
Obvious.
Corollary 3.3:
Let N be a nearfield space such that Nil (N) is a divided prime ideal nearfield space of N, and let Q be a divided prime of ideal of N such that Q
≠
Nil (N). Then Q is branched if and only if Rad (J) Then Q is branched iff Rad (j) = Q for some ideal J
≠
Q of nearfield space N.
Proposition 3.4:
Let N be a nearfield space such that
Nil
(
N
) is a divided prime ideal of N. Suppose that J is a proper ideal of N such that J contains a nonnilpotent subnearfield space of N and for some N
≥
1, J
n
is a divided ideal of nearfield space N for each n
≥
N. Then Q =
1
≥
nn
J
is a divided prime ideal of nearfield space N.
Proof:
Obvious.
In view of the above proposition, we have the following corollary.
Corollary 3.5:
Let N be a nearfield space such that
Nil
(N) is a divided prime ideal of N, and let J be proper ideal of N such that J contains a nonnilpotent of N. Then the following statements are equivalent: (i)
J
n
= J
m
for some positive integers n
≠
m and J
n
is a divided ideal of N. (ii)
J is a divided prime ideal of N and J = J
2
.
Proof: Obvious
The following result follows directly from the definition of strongly prime ideal and a quasilocal nearfield space with maximal ideal M is a PVNFS if and only if M is strongly prime.
Proposition 3.6:
Let N be a nearfield space such that Nil (N) is a divided prime ideal of N, and let J be a proper ideal of N such that J contains a nonnilpotent of N. Then the following statements are equivalent: (i)
N is
φ
PVNFS (ii) bM is a divided ideal of N for each b
∈
N \ Nil (N).
Proof:
Obvious An element d in a nearfield space N is called a proper divisor of s
∈
N if
s = dm
for some nonunit m
∈
N.
Proposition 3.7:
For a quasilocal nearfield space N with maximal ideal M, the following statements are equivalent: (i)
N is a PVNFS; bM is a divided ideal for each b
∈
M.
N V Nagendram* / A Note On Divided NearField Spaces and Pseudo – Valuation NearField Spaces Over Regular NearRings (Dnf
PvnfsO
Nr) / IJMA 6(4), April2015. © 2015, IJMA. All Rights Reserved 34
Proof:
Obvious
In the following proposition we make connection between
φ
PVNFS’s and PVNFS’s.
Proposition 3.8:
A nearfield space N is a
φ
PVNFS
⇔
Nil
(N) is a divided prime ideal of N and
∀
a, b
∈
N \
Nil
(N), either b a
∈
N or d  b
∈
N for each proper divisor d of a.
Proof:
Obvious
Proposition 3.9:
A nearfield space N is a
φ
PVNFS
⇔
Nil
(N) is divided prime ideal of N and N/
Nil
(N) is a PVNFS.
Proof:
Obvious.
SECTION 4: MAIN RESULTS ON DIVIDED NEARFIELD SPACES, PVNFS AND CPVNFS
In this section, let a valuation domain VD and VNFS valuation nearfield space and chained nearfield space CNFS. We then have the following implications, none of which are reversible. V D
⇒
PVD
⇒
VNFS
⇒
PVNFS
⇒
φ
 PVNFS and VD
⇒
CNFS
⇒
φ
 CNFS
⇒
φ
 PVNFS. We start with the following lemma.
Lemma 4.1:
Let N be a
φ
 PVNFS, and let Q be a prime ideal of N. then x
1
Q
⊂
Q for each x
∈
T(N) \ N.
Proof:
Obvious.
Proposition 4.2:
Let N be a
φ
 PVNFS and z
∈
T(N) \ N be integral over N. Then there is a minimal monic polynomial
f
(x)
∈
N[x] such that
f(x) = 0
and all nonzero coefficients of
f(x)
are units in N. Furthermore, if
g(x)
is a minimal monic polynomial in N[x] such that
g(x)
=0, then
g(0)
is a unit in N.
Proof:
Obvious It is wellknown ([15],[1],[4],[7]) that the integral closure of a PVNFS is a PVNFS. In view of the above result, one can give replica proof of this fact. For a nearfield space N, let N
′
denotes the integral closure nearfield space of N inT (N).
Proposition 4.3:
Let N be a
φ
 PVNFS with maximal ideal M, and let A be a over nearfield space of N such that A
⊂
N
′
. Then A is a
φ
 PVNFS with maximal ideal M.
Proof:
Obvious
Proposition 4.4:
Let N be a
φ
 PVNFS with maximal ideal M, and Let A be a overnearfield space of N. Then the following statements are equivalent: (i)
A = A
Q
is a
φ
 CNFS for some nonmaximal prime ideal Q of N (ii)
IA = A for some proper ideal J of N (iii)
1/s
∈
A for some nonzero divisor s
∈
M.
Proof:
Obvious
Corollary 4.5
[6, theorem 3]
:
Let N be a
φ
 PVNFS with maximal ideal M, and let A be a overnearfield space of N such that A is a
φ
 CNFS with maximal ideal N. If Q = N
∩
K
≠
M, then A = N
Q
.
Proof:
is obvious.
Proposition 4.6:
Let N be a
φ
 PVNFS with maximal ideal M and u
∈
(M: M)\N. Then N (v) is a
φ
 PVNFS if and only if N (v) is quasilocal nearfield space. Furthermore, if N (v) is quasilocal nearfield space for some u
∈
(M: M)\N, then N (v) is a
φ
 PVNFS with maximal ideal M.
Proof:
is obvious.
Corollary 4.7:
Let N be a
φ
 PVNFS with maximal ideal M. If C is a overnearfield space of N such that C does not contain an element of the forth 1/s for some nonzero divisor s
∈
M, then C
⊂
(M: M).
N V Nagendram* / A Note On Divided NearField Spaces and Pseudo – Valuation NearField Spaces Over Regular NearRings (Dnf
PvnfsO
Nr) / IJMA 6(4), April2015. © 2015, IJMA. All Rights Reserved 35
Proof:
is obvious.
Corollary 4.8:
Let N be a
φ
 PVNFS with maximal ideal M. Then every overnearfield space C of N is a
φ
 PVNFS
iff N (v)
is quasilocal nearfield space for each u
∈
(M: M)\N.
Proof:
Obvious
Note 4.9:
A nearfield space N is
φ
 CPVNFS
if and only if
Nil(N)
is a divided prime ideal of N and
∀
a, b
∈
N\
Nil(N)
, either ab
∈
N or ba
∈
N. We have the following result which is a generalization of [6, proposition 6].
Proposition 4.10:
Let N be a
φ
 PVNFS. Then (i) N is a Nagendram nearfieldspace. (ii) If N
≠
T (N), then T(N) is
φ
 CPVNFS.
Proof: To prove (i):
Since Z(N) is a prime ideal of N by property 1.6, N has few zero divisors. Hence, N is a Nagendram nearfield space by [16, theorem 7.2]. Proved (i).
To prove (ii):
Since
Nil(N)
is a divided prime ideal of N,
Nil(T(N)) = Nil(N)
. Now let x, y
∈
T(N) \
Nil(N)
. Then x = a/s and y = b/s
∀
a, b
∈
N \
Nil(N)
and s
∈
N \ Z(N). by note 4.9, we need to show that either xy in T(N) or yx in T(N). if ab in N, then xy in T(N). Hence assume that ab in N. Since, N is a
φ
 PVNFS and N
≠
T(N),
b

ad
in N for some d
∈
M \ Z(N). Thus,
ad = bc
for some c
∈
N. Thus, a/s = (b/s)(c/d). Thus, y x in T(N). Proved (ii). Therefore, this completes the proof of proposition.
Remark 4.11:
Let N be a
φ
 PVNFS with maximal ideal M such that M contains a nonzero divisor of N, and J be a proper ideal of N. Since U =(M : M) is a
φ
 CPVNFS with maximal ideal M of N, it is easy to see that there exists a
φ
 CPVNFS U between N and T(N) such that IU
≠
U.
Theorem 4.12:
Let N be a
φ
 PVNFS with maximal ideal M such that M contains a nonzero divisor of N, let C be a over nearfield space of N (N
⊂
C
⊂
T(N)), and let J be a proper ideal of C. Then there exists a
φ
 CPVNFS A such that C
⊂
A
⊂
T(N) and JA
≠
A.
Proof:
Obvious.
Proposition 4.13:
Let N be a
φ
 PVNFS and be a over nearfield space of N such that A is a
φ
 CPVNFS. Then, N
′
⊂
A.
Proof:
we prove this in the way of Negative proof. Then there is an x
∈
N
′
\A. Hence, since N
′
is a
φ
 PVNFS with maximal ideal M by proposition 4.3, x is a unit in N
′
. Since x
∉
A and A is a
φ
 CPVNFS, x
1
∈
A. Since, x
∈
N
′
, x
∈
N[x
1
] by [17, theorem 15]. Hence, x
∈
N[x
1
]
⊂
A, which is a contradiction, thus N
′
⊂
A. This completes the proof of the proposition.
Theorem 4.14:
Let N be a
φ
 PVNFS with maximal ideal M such that M contains a nonzero divisor. Then N
′
is the intersection of all the
φ
 CPVNFs between N and T(N).
Proof:
By proposition 4.13, N
′
is contained in the intersection of all the
φ
 CPVNF between N and T(N). Let y
∈
the intersection of all the
φ
 CPVNFs between N and T(N). we must show that y
∈
N
′
. Suppose not. By [17, theorem 15], y
∉
C = N[y
1
]. Let J = y
1
C. Then J is a proper ideal of C. by theorem 4.12 there is a
φ
 CPVNF A between C and T(N) such that JA
≠
A. But by hypothesis y
∈
A, and we have our contradiction. This completes the proof of the theorem.
Theorem 4.15
[4, Th. 15(1)]
:
Let N be a
φ
 PVNFS with maximal ideal M. Then every overnearfield space of N is a
φ
 PVNFS
⇔
every
φ
 CNFS between N and T(N) other than (M:M) is of the form N
Q
for some nonmaximal prime ideal Q of N.
Proof:
⇒
(if)
If T(N) = N, then there is nothing to prove. Hence, assume that M contains a nonzero divisor of N. Suppose that every over nearfield space of N is a
φ
 PVNFS. Then N
′
= (M: M). Let C be a over nearfield space of N such that C
≠
(M : M) and C is
φ
 CNFS. Since every over nearfield space of N not contained in N
′
= (M: M) by proposition 4.7 and hence is a
φ
 PVNFS with maximal ideal M by proposition 4.3 and (M : M) is the only
φ
 CNFS