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International Journal of Algebra, Vol. 8, 2014, no. 10, 451 - 458
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ija.2014.4353
An Element Prime to and Primary to
Another Element in Lattice Modules
C. S. Manjarekar
Department of Mathematics
Shivaji University
Kolhapur (India)-416004
A. V. Bingi
Department of Mathematics
St. Xavier’s College (Autonomous)
Mumbai (India)-400001
c 2014 C. S. Manjarekar and A. V. Bingi. This is an open access article
Copyright distributed under the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Abstract
In this paper, we introduce the concepts of “An element prime to and
primary to another element in lattice modules”.Using these concepts,we
generalize some results proved by C.S.Manjarekar and N.S.Chavan[4].
Some characterizations of an element prime to and primary to another
element in lattice modules are obtained.It is shown that if a family
of elements is primary to an element A then their supremum is also
primary to A and if ∧Bα is primary to an element A then each Bα is
α
primary to A.Finally,the characterization of an element non-primary to
another element is obtained.
Mathematics Subject Classification: 06F10, 06F05, 13A15
452
C. S. Manjarekar and A. V. Bingi
Keywords: Multiplicative lattice, lattice module, prime element, primary
element, prime to another element, primary to another element, minimal prime
1
Introduction
A multiplicative lattice L is a complete lattice provided with commutative,
associative and join distributive multiplication in which the largest element
1 acts as a multiplicative identity. An element a ∈ L is called compact if
for X ⊆ L, a 6 ∨X implies the existence of a finite number of elements
a1 , a2 , ..., an in X such that a 6 a1 ∨ a2 ∨ ... ∨ an . The set of compact elements
of L will be denoted by L∗ .An element a ∈ L is said to be proper if a < 1. A
proper element p ∈ L is called prime element if ab 6 p implies a 6 p or b 6 p
where a, b ∈ L and is called primary element if ab 6 p implies a 6 p or bn 6 p
for some n ∈ Z+ where a, b ∈ L∗ .
Throughout this paper, L denotes a compactly generated multiplicative
lattice with 1 compact in which every finite product of compact elements is
compact. For a, b ∈ L,(a : b) = ∨{x ∈ L | xb 6 a}. The radical of a ∈ L
√
is denoted by a and is defined as ∨{x ∈ L∗ | xn 6 a, f or some n ∈ Z+ }.
√
An element a ∈ L is called semiprimary if a is a prime element. An element
√
a ∈ L is called semiprime if a = a. An element a ∈ L is called meet
prime if a1 ∧ a2 6 a implies a1 6 a or a2 6 a and is called completely meet
prime if ∧aα 6 a implies aα 6 a for some α. An element a ∈ L is called
α
√
meet semiprimary if a is a meet prime element and is called completely meet
√
semiprimary if a is completely meet prime element. A prime element p ∈ L is
called minimal prime if there exists no other prime element q such that q 6 p.
For all these definitions in multiplicative lattices, one can refer[2].
Let M be a complete lattice and L be a multiplicative lattice. Then M is
called L-module or module over L if there is a multiplication between elements
of L and M written as aB where a ∈ L and B ∈ M which satisfies the following
1 (∨aα )A = ∨aα A , 2 a(∨Aα ) = ∨a Aα , 3 (ab)A = a(bA) , 4
properties:- α
α
α
α
5 0A = OM , f or all a, aα , b ∈ L and A, Aα ∈ M , where 1 is the
1A = A , supremum of L and 0 is the infimum of L.We denote by OM and IM for the
least element and the greatest element of M respectively. Elements of L will
generally be denoted by a, b, c, ... and elements of M will generally be denoted
by A,B,C,...
Let M be an L-module. If N ∈ M and a ∈ L then (N : a) = ∨{X ∈ M |
453
Lattice modules
aX 6 N }. If A, B ∈ M , then (A : B) = ∨{x ∈ L | xB 6 A}. An L-module
M is called a multiplication lattice module if for every element N ∈ M there
exists an element a ∈ L such that N = aIM . In this paper, a lattice module
M will be a multiplication lattice module.An element N ∈ M is said to be
proper if N < IM . A proper element N of M is said to be prime if aX 6 N
implies X 6 N or aIM 6 N ,that is a 6 (N : IM ) for every a ∈ L , X ∈ M .
If N is a prime element of M then (N : IM ) is prime element of L. An element
N < IM in M is said to be primary if aX 6 N implies X 6 N or an IM 6 N
that is an 6 (N : IM ) for some n ∈ Z+ .An element N of M is called a radical
p
p
element if (N : IM ) = (N : IM ) where (N : IM ) = ∨{x ∈ L∗ | xn 6 (N :
IM ), f or some n ∈ Z+ } = ∨{x ∈ L∗ | xn IM 6 N, f or some n ∈ Z+ }.
p
An element A ∈ M is called meet semiprimary if (A : IM ) is a meet prime
p
element and A ∈ M is called completely meet semiprimary if (A : IM ) is
completely meet prime element. The following result will be used wherever
necessary:- A : (∨Bα ) = ∧(A : Bα ) where A,Bα ∈ M . For all the definitions
α
α
in lattice Module and some other definitions, one can refer[3].
2
An element prime to and primary to another element
We introduce the concepts of an element prime to and an element primary
to another element in lattice modules.This is a generalization of the concept
introduced by C.S.Manjarekar and N.S.Chavan [4] in multiplicative lattices.
Definition:- An element B ∈ M is said to be primary to an element A ∈ M
if for all x ∈ L∗ , xB 6 A implies xn IM 6 A for some n ∈ Z+ . An element
B ∈ M is said to be prime to an element A ∈ M if for all x ∈ L , xB 6 A
implies xIM 6 A.
We note that, if B is prime to A then B is primary to A.But the converse is
not true which is shown in the following example.
Example:- Consider the lattice L of ideals of ring R=< Z1 2 , + , . >.
Then the only ideals of ring R are the principal ideals (0),(1),(2),(3),(4),(6).
L={(0),(1),(2),(3), (4),(6)} is a multiplicative lattice and L is a module over
itself. Its lattice structure is as follows:-
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C. S. Manjarekar and A. V. Bingi
From the above multiplication table, we see that element b is primary to a but
not prime to a.
Theorem 2.1. No proper non-zero element of M is primary to itself.
Proof. Suppose A is a proper non-zero element of M and A is primary to itself.
Since 1A=A,we have,1IM 6 A.This gives A = IM ,a contradiction.
The element prime to another element is obviously primary to that element.
Under which conditions, the converse holds is proved in the next result.
Theorem 2.2. Let A ∈ M be a radical element. Then B ∈ M is primary to
A if and only if B is prime to A.
Proof. Assume that B is primary to A in M. Let xB 6 A where x ∈ L∗ . Then
√
xn IM 6 A, f or some n ∈ Z+ . Consequently, x 6 A : IM = A : IM and hence
xIM 6 A. The converse is obvious.
Theorem 2.3. Let A, B ∈ M . Then B is primary to A if and only if A : B 6
√
A : IM .
Proof. Assume that B is primary to A in M. Let x ∈ L∗ be such that x 6
A : B. Then xB 6 A and by hypothesis,xn IM 6 A, f or some n ∈ Z+ .
√
√
Hence, x 6 A : IM . So we have,A : B 6 A : IM . Conversely, assume that
√
√
A : B 6 A : IM . Let x ∈ L∗ be such that xB 6 A. Then, x 6 A : IM
which implies that xn IM 6 A, f or some n ∈ Z+ .
Lattice modules
455
We now obtain a characterization of an element prime to another element.
Theorem 2.4. Let A, B ∈ M . Then B is prime to A if and only if A : B =
A : IM .
Proof. Assume that B is prime to A in M. Let x ∈ L be such that x 6 A : B.
Then xB 6 A and by hypothesis,xIM 6 A. This shows that A : B 6 A : IM .
On the other hand, let x 6 A : IM . Then xB 6 xIM 6 A.So,x 6 A : B and we
conclude that A : B = A : IM . Conversely, assume that A : B = A : IM .Let
x ∈ L be such that xB 6 A. Then, x 6 A : IM . Hence, B is prime to A.
Next we have a characterization of an element primary to another element
and non-prime to it.
Theorem 2.5. Let A, B ∈ M . Then B is primary to A but non-prime to A if
√
and only if A : IM < A : B 6 A : IM .
Proof. Assume that B is primary to A but non-prime to A. Then by theorems
√
(2.3) and (2.4),A : B 6 A : IM but A : B 6= A : IM . Since A : IM 6 A : B ,
√
we must have, A : IM < A : B 6 A : IM . Conversely assume that A : IM <
√
A : B 6 A : IM . Then by theorems (2.3) and (2.4),it follows that B is
primary to A but non-prime to A.
Theorem 2.6. Let A, B ∈ M . If A is a prime element and B A then B is
prime to A.
Proof. Let x ∈ L be such that xB 6 A. Since B A and A is a prime
element, it follows that x 6 A : IM or equivalently xIM 6 A. Hence, B is
prime to A.
Theorem 2.7. Let A, B ∈ M . Then B is prime to A if and only if B is prime
to A : x for every x ∈ L.
Proof. Assume that B is prime to A in M and let yB 6 A : x where y ∈ L.
Then xyB 6 A. As B is prime to A, we have, xyIM 6 A. Consequently,
yIM 6 A : x. Thus, B is prime to A : x for every x ∈ L. The converse is
obvious.
The previous theorem leads us to a following result.
Corollary 2.1. Let A, B ∈ M . If B is primary to A and A is a radical element
then B is primary to A : x for every x ∈ L.
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C. S. Manjarekar and A. V. Bingi
Proof. The proof follows from the theorems (2.2) and (2.7).
Theorem 2.8. Let A, B ∈ M . Then B is primary to A if and only if X > B
is primary to A where X ∈ M .
Proof. Assume that B is primary to A in M and let X > B where X ∈ M . If
yX 6 A where y ∈ L∗ then yB 6 A. By hypothesis, y n IM 6 A, f or some n ∈
Z+ .This shows that X > B is primary to A. The converse is obvious.
The above theorem leads us to the following two results; the proofs of which
are obvious.
Corollary 2.2. If {Bα } is a family of elements primary to A in M then ∨Bα
α
is primary to A.
Corollary 2.3. If ∧Bα is primary to A in M then each Bα is primary to A
α
where {Bα } ⊆ M .
Now, we obtain a characterization of an element non-primary to another
element.
Theorem 2.9. Let A, B ∈ M . Then B is non-primary to A if and only if
each X 6 B is non-primary to A where X ∈ M .
Proof. Assume that B is non-primary to A in M. Then by theorem(2.3), we
√
have A : B A : IM . Let X ∈ M be such that X 6 B. Then A : B 6 A : X
√
implies A : X A : IM . Again by theorem(2.3),X 6 B is non-primary to
A. The converse is obvious.
The above theorem leads us to the following two obvious corollaries.
Corollary 2.4. If {Bα } is a family of elements non-primary to A in M then
∧Bα is non-primary to A.
α
Corollary 2.5. If ∨Bα is non-primary to A in M then each Bα is non-primary
α
to A where {Bα } ⊆ M .
Theorem 2.10. Let a, b ∈ L. If bIM is an element of M and abIM is nonprimary to A ∈ M then either aIM is non-primary to A or bn IM is non-primary
to A for some n ∈ Z+ .
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Lattice modules
Proof. Assume that abIM is non-primary to A ∈ M where a, b ∈ L. Suppose,
if possible,aIM is primary to A and bn IM is primary to A for all n ∈ Z+ . Let
c ∈ L∗ be such that cabIM 6 A. As aIM is primary to A,(bc)n IM 6 A for some
n ∈ Z+ . As bn IM is primary to A,we have,(cn )k IM 6 A for some k ∈ Z+ . This
contradicts the fact that abIM is non-primary to A which proves the result.
Theorem 2.11. Suppose A ∈ M is completely meet semiprimary element. Let
{Bα } ⊆ M . If each Bα is non-primary to A then ∨Bα is non-primary to A.
α
Proof. As each Bα is non-primary to A ∈ M it follows by theorem(2.3) that
√
each A : Bα A : IM . Since, A is completely meet semiprimary element of
√
M, we have,∧(A : Bα ) = A : (∨Bα ) A : IM . Again, by theorem(2.3), ∨Bα
α
α
α
is non-primary to A.
We denote by QA the element ∨{X ∈ M | X is non − primary to A}. In
the next result, we prove the property of the element QA .
Theorem 2.12. If A ∈ M is completely meet semiprimary element then QA
is a primary element and is non-primary to A.
Proof. By theorem(2.11),QA is non-primary to A. Let aN 6 QA where N ∈ M
and a ∈ L. Since,M is a multiplication lattice module, an element N ∈ M can
be written as N = bIM for some b ∈ L and we have abIM 6 QA . So,abIM is
non-primary to A. By theorem(2.10), either bIM is non-primary to A or an IM
is non-primary to A for some n ∈ Z+ . Hence, bIM 6 QA or an IM 6 QA .
Thus,N 6 QA or an 6 (QA : IM ) for some n ∈ Z+ .So,QA is a primary element
of M.
Theorem 2.13. Assume that every finite product of compact elements in L is
a compact element in L. Let A ∈ M be completely meet semiprimary element.
√
If every prime element of M is a minimal prime element then QA : IM =
√
A : IM .
√
Proof. By theorem(2.12),QA is a primary element of M. Then QA : IM is
a prime element of L. If A = IM then QA = ∨{X ∈ M | X is non −
√
primary to IM }=∨{X ∈ M | IM : X IM : IM = 1}. Since IM :
X = 1 ,we must have, A < IM . By theorem(2.1),A is non-primary to A
√
√
and hence A 6 QA . Thus, A : IM 6 QA : IM . On the other hand, let
√
x be a compact element of L such that x A : IM . Define Φ = {xn y |
√
y QA : IM , y is compact , n ≥ 0}. Let xn y, xm z ∈ Φ for some compact
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C. S. Manjarekar and A. V. Bingi
√
√
elements y, z in L such that y, z QA : IM for some n, m ≥ 0. As QA : IM
√
is a prime element of L,yz QA : IM . Also, the product of compact elements
is compact. Consequently,(xn y)(xm z) = xn+m yz ∈ Φ. Suppose 0∈ Φ. Then
√
0=xn y for some compact element y QA : IM , n ≥ 0. As y (QA : IM ),
yIM is primary to A. We have 0=xn y 6 (A : IM ). So,xn yIM 6 A. Hence
√
(xn )m IM 6 A. This shows that x 6 A : IM which contradicts the fact that
√
x A : IM . Thus Φ is a multiplicatively closed subset of L∗ such that 0∈
/ Φ.
By separation lemma[1], there exists a prime element p such that t p for all
√
√
t ∈ Φ. Now we show that p 6 QA : IM .Suppose p QA : IM .Then there
√
is a compact element b in L such that b 6 p but b QA : IM . So, xn b ∈ Φ
for n ≥ 0.As t p for all t ∈ Φ, we have, xn b p for n ≥ 0. But then b p
√
which is a contradiction. Hence,we must have p 6 QA : IM . Therefore the
√
hypothesis implies that p = QA : IM . Now, by definition of Φ, x ∈ Φ. Then
√
√
√
x p = QA : IM . Thus, x A : IM gives x QA : IM . This shows that
√
√
QA : IM 6 A : IM . This completes the proof.
References
[1] Anderson D D, Abstract commutative ideal theory without chain conditions, Algebra Universalis, 6(1976),131-145.
[2] Alarcon F, Anderson D D and Jayaram C, Some results in abstract commutative ideal theory, Periodica Math.Hung., 30(1995), 1-26.
[3] Callialp F and Tekir U, Multiplication lattice modules, Iranian Journal of
Science and Technology, IJST (2011), A4, 309 - 313.
[4] Manjarekar C S and Chavan N S, An element primary to another element,
Journal of the Indian Math.Soc., Vol-71,Nos. 1-4(2004),55-60.
Received: March 1, 2014