A NOTE ON DIVIDED NEAR-FIELD SPACES AND φ-PSEUDO – VALUATION NEAR-FIELD SPACES OVER REGULAR δ-NEAR-RINGS (DNF-φ PVNFS-O-δ-NR

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  International Journal of Mathematical Archive-6(4), 2015, 31-38   Available online through www.ijma.info   ISSN 2229 – 5046   International Journal of Mathematical Archive- 6(4), April – 2015 31    A NOTE ON DIVIDED NEAR-FIELD SPACES AND -PSEUDO – VALUATION NEAR-FIELD SPACES OVER REGULAR -NEAR-RINGS (DNF-   PVNFS-O-   -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: 18-02-15; Revised & Accepted On: 23-04-15)    ABSTRACT  L et N be a commutative near-field space with 1 and T(N) be the total quotient near-field space such that Nil(N) is a divided prime ideal of a near-field space N. Then N is called a φ  -chained near-field 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 near-field space ( φ  -PVNF) if for every x, y ∈   N \ Nil (N) either x |   y or y |   xm where for each non-unit element m ∈   N. We show that a near-field space N is a φ  -PVNF iff Nil (N) is a divided prime ideal and N/Nil(N) is a pseudo-valuation domain. Also, we show that every over near-field space of a Quasi-local near-field space N with maximal ideal M is a φ  -PVNF iff N (v) for each v ∈   (M: M) \  N iff every over- near-field space of N is a quasi-local iff every φ  -CNF between N and T(N) other than N and (M : M) is of the form N   p  for some non-maximal prime ideal P of N. Among other results, we show that if A is an over-near-field 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 near-field 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:  Near-field, Near–field space, total quotient near-field space, divided prime ideal, quasi near-field space, Quasi-local near-field space, over-near-field space, pseudo –valuation near-field space, maximal ideal, maximal ideal. SECTION 1: INTRODUCTION Throughout this paper, N denotes as Near-field space has zero symmetric near-ring with identity. We begin by recalling some background material. With reference to ([1], [4]) the author generalized the study of pseudo-valuation domains to the context of extending to arbitrary near-field spaces possibly with non-zero zero divisors. For a near-field space N with total quotient near-field 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 near-field homomorphism from T(N) into K, and φ  is restricted to near-field space N is also a near-field 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 quasi-local near-field space N with maximal ideal M is a φ -PVNFS if and only if N(v) is a quasi-local near-field space for each v ∈  (  M   :  M  ) \ N if and only if every over-near-field space of N is quasi-local near-field space if and only if every over-near-field space contained in (  M   :  M  ) is quasi-local near-field 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 non-maximal prime ideal P of N. Among the other results, we show that if A is an over-near-field 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 near-field 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 near-field space. Then T(N) denote the total quotient near-field space of a near-field space N.  Nil (N) denotes the near-field 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 Near-Field Spaces and -Pseudo – Valuation Near-Field Spaces Over Regular -Near-Rings (Dnf-   Pvnfs-O-   -Nr) / IJMA- 6(4), April-2015. © 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 near-field space and hence is quasi-local near-field space. Property 1.2:  A φ -PVNFS is a divided near-field space and hence is quasi-local near-field space. Property 1.3:  A sub–near-field space is a PVNFS iff   it is a φ -PVNFS iff   it is a PVD. Property 1.4:  A near-field space N is a PVNFS if and only if ∀  a, b ∈  N, either a/b ∈  N or b/a ∈  N for each non-unit c ∈  N. Property 1.5:  A near-field 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 ∀  non-unit 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 near-field space N. SECTION 2: PRELIMINARY RESULTS AND EXAMPLES Definition 2.1:  A near-field space N, with quotient near-field space K of N is called a pseudo-valuation domain [PVD] near-field 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 near-field space N is said to be strongly prime in N if aP and bP are comparable under inclusion of near-field spaces ∀  a, b ∈  N. Definition 2.4:  A near-field space N is called a pseudo-valuation near-field space (PVNFS) if each prime ideal of N is strongly prime. A PVNFS is necessarily quasi-local near-field space. Note 2.5:  A near-field space is a pseudo-valuation near-field space (PVNFS) if and only if it is pseudo-valuation domain [PVD]. Definition 2.6:  A prime ideal P of a near-field space N is called divided if it is comparable under inclusion to every ideal of near-field space N. Definition 2.7:  A near-field space N is called a divided near-field space if every prime ideal of a near-field space N is divided. Definition 2.8:  A prime ideal Q of φ (N) is called a K-strongly 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 K-strongly prime, then φ (N) is called a K-pseudo-valuation near-field space (K-PVNFS). Definition 2.10:  A prime ideal P of near-field space N is called a φ -strongly prime ideal if φ (P) is a K-strongly 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 K-strongly prime ideal of φ (N). If each prime ideal of near-filed space N is φ -strongly prime, then N is called a φ - pseudo-valuation near-field space ( φ -PVNFS). Definition 2.12:  a near-field space N is called a φ -chained near-field 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 near-field space ( φ -CNFS) is a division near-field space and hence is quasi-local near-field space. Hence, ∀  n ≥  0 ∃  a φ -CNFS of krull dimension n that is not a chained near-field space. Definition 2.14:  A proper ideal of a near-field 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 near-field space. Definition 2.15:  A prime ideal Q of a near-field space B is branched if Rad (J) = Q for some primary ideal J ≠  Q of B.  N V Nagendram* / A Note On Divided Near-Field Spaces and -Pseudo – Valuation Near-Field Spaces Over Regular -Near-Rings (Dnf-   Pvnfs-O-   -Nr) / IJMA- 6(4), April-2015. © 2015, IJMA. All Rights Reserved 33 Note 2.16:  A prime ideal Q of a near-field 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 near-field spaces. Definition 2.17:  An ideal of a near-field space N is called regular if it contains a non-zero divisor of N. If every regular ideal of N is generated by its set of non-zero divisors, then N is called as Nagendram near-field space. Definition 2.18:  A near-field space N has few zero-divisors if Z(N) is a finite union of prime ideals. SECTION 3: RESULTS ON DIVIDED NEAR-FIELD 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 near-field space N is a divided near-fields domain. Hence, I state the following result without proof. Proposition 3.1:  [5, Proposition 2.1] Let D be division near-field space domain with maximal ideal K and krull dimension n, say K = Q n   ⊃  Q n-1   ⊃  Q n-2 ⊃  ……… ⊃  Q 1   ⊃  {0}, where the Q  j s are the distinct prime ideals of division near-field 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 near-field space domain D and rad (I) = Q  j  . (ii)   N:= D/I is a divided near-field 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 near-field 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 near-field space N. Proof: Obvious.  Corollary 3.3: Let N be a near-field space such that Nil (N) is a divided prime ideal near-field 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 near-field space N. Proposition 3.4: Let N be a near-field 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 non-nilpotent sub-near-field space of N and for some N ≥  1, J n  is a divided ideal of near-field space N for each n ≥ N. Then Q =  1 ≥ nn  J   is a divided prime ideal of near-field space N. Proof: Obvious.   In view of the above proposition, we have the following corollary. Corollary 3.5: Let N be a near-field 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 non-nilpotent 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 quasi-local near-field space with maximal ideal M is a PVNFS if and only if M is strongly prime. Proposition 3.6: Let N be a near-field 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 non-nilpotent 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 near-field space N is called a proper divisor of s ∈  N if s = dm for some non-unit m ∈  N. Proposition 3.7: For a quasi-local near-field 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 Near-Field Spaces and -Pseudo – Valuation Near-Field Spaces Over Regular -Near-Rings (Dnf-   Pvnfs-O-   -Nr) / IJMA- 6(4), April-2015. © 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 near-field 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 near-field 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 NEAR-FIELD SPACES, -PVNFS AND -CPVNFS In this section, let a valuation domain VD and VNFS valuation near-field space and chained near-field 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 non-zero 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 well-known ([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 near-field space N, let N ′  denotes the integral closure near-field space of N inT (N). Proposition 4.3:  Let N be a φ  - PVNFS with maximal ideal M, and let A be a over near-field 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 over-near-field space of N. Then the following statements are equivalent: (i)   A = A Q  is a φ  - CNFS for some non-maximal prime ideal Q of N (ii)   IA = A for some proper ideal J of N (iii)   1/s ∈  A for some non-zero 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 over-near-field 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 quasi-local near-field space. Furthermore, if N (v) is quasi-local near-field 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 over-near-field space of N such that C does not contain an element of the forth 1/s for some non-zero divisor s ∈  M, then C ⊂  (M: M).  N V Nagendram* / A Note On Divided Near-Field Spaces and -Pseudo – Valuation Near-Field Spaces Over Regular -Near-Rings (Dnf-   Pvnfs-O-   -Nr) / IJMA- 6(4), April-2015. © 2015, IJMA. All Rights Reserved 35 Proof: is obvious.  Corollary 4.8:  Let N be a φ  - PVNFS with maximal ideal M. Then every over-near-field space C of N is a φ  - PVNFS iff N (v) is quasi-local near-field space for each u ∈  (M: M)\N. Proof: Obvious Note 4.9: A near-field 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 a|b ∈  N or b|a ∈  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 near-field-space. (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 near-field 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 x|y in T(N) or y|x in T(N). if a|b in N, then x|y in T(N). Hence assume that a|b 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 non-zero 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 non-zero divisor of N, let C be a over near-field 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 near-field 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 non-zero 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 over-near-field 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 non-maximal prime ideal Q of N. Proof:   ⇒ (if)   If T(N) = N, then there is nothing to prove. Hence, assume that M contains a non-zero divisor of N. Suppose that every over near-field space of N is a φ  - PVNFS. Then N ′  = (M: M). Let C be a over near-field space of N such that C ≠  (M : M) and C is φ  - CNFS. Since every over near-field 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
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