Second and Secondary Lattice Modules

Let M be a lattice module over the multiplicative lattice L. A nonzero L-lattice module M is called second if for each a ∈ L, a1M = 1M or a1M = 0M. A nonzero L-lattice module M is called secondary if for each a ∈ L, a1M = 1M or a n1M = 0M for some n > 0. Our objective is to investigative properties of second and secondary lattice modules.

A multiplicative lattice is a complete lattice in which there is defined as a commutative, associative multiplication which distributes over arbitrary joins and has the compact greatest element 1 (least element 0 ) as a multiplicative identity (zero). An element ∈ is said to be proper if < 1 . An element < 1 in is said to be prime if ≤ implies either ≤ or ≤ . If 0 is prime, then is said to be a domain. For ∈ , we define √ = ⋁{ ∈ : ≤ for some integer }. An element < 1 in is said to be primary if ≤ implies either ≤ or ≤ √ .
If , belong to , ( : ) is the join of all ∈ such that ≤ . An element of is called meet principal if ⋀ = (( : ) ⋀ ) for all , ∈ . An element of is called join principal if (( ⋁ ): ) = ⋁( : ) for all , ∈ . ∈ is said to be principal if is both meet principal and join principal. ∈ is said to be weak meet (join) principal if ⋀ = ( : ) ( ⋁(0 : ) = ( : )) for all ∈ . An element of a multiplicative lattice is called compact if ≤ ⋁ implies ≤ 1 ⋁ 2 ⋁ ⋅ ⋅ ⋅ ⋁ for some subsets { 1 , 2 , . . . , }. If each element of is a join of principal (compact) elements of , then is called a lattice (CG-lattice). If is a -lattice and is a primary element, then √ is prime [1, Lemma 2.1].
Let be a complete lattice. Recall that is a lattice module over the multiplicative lattice or simply anmodule in case there is a multiplication between elements of and , denoted by for ∈ and ∈ , which satisfies the following properties: (1) ( ) = ( ); (2) (⋁ )(⋁ ) = ⋁ , ; (3) 1 = ; (4) 0 = 0 ; for all , , in and for all , in . Let be an -module. If , belong to , ( : ) is the join of all ∈ such that ≤ . Particularly, (0 : 1 ) is denoted by ( ). If ∈ and ∈ , then ( : ) is the join of all ∈ such that ≤ . An element of is called meet principal if ( ∧ ( : )) = ∧ for all ∈ and for all ∈ . An element of is called join principal if ∨ ( : ) = (( ∨ ): ) for all ∈ and for all ∈ . is said to be principal if it is both meet principal and join principal. In special cases, an element of is called weak meet principal (weak join principal) if ( : ) = ∧ (( : ) = ∨ (0 : )) for all ∈ (for all ∈ ). is said to be weak principal if is both weak meet principal and weak join principal.
Let be an -module. An element in is called compact if ≤ ⋁ implies ≤ 1 ∨ 2 ∨ ⋅ ⋅ ⋅ ∨ for some subsets { 1 , 2 , . . . , }. The greatest element of will be denoted by 1 . If each element of is a join of principal (compact) elements of , then is called a lattice module ( -lattice module).
Let be an -module. An element ∈ is said to be proper if < 1 . For all elements of , [ , 1 ] is a set of 2 The Scientific World Journal all ∈ such that ≤ ≤ 1 and [ , 1 ] is an -lattice module with ⋅ = ∨ for all ∈ and ∈ such that ≤ .
Example 3. Let be the integers, let be the rational numbers, and let be -module. Suppose = ( ) is the set of all ideals of and = ( ) is the set of all submodules of . Thus, as -lattice module is a second module, since for every integer ∈ , ( ) = or ( ) = 0.

Remark 4. Every second lattice module is a secondary lattice.
But the converse is not true. For this, we can give the following example.
Example 5. Let be the integers and let 4 be -module. Suppose that = ( ) is the set of all ideals of and = ( 4 ) is the set of all submodules of 4 . Thus, as -lattice module is a secondary lattice module, which is not a second lattice module.
Example 6. Let be the integers and = ( ) the set of all ideals of . Thus, as -lattice module is neither a second lattice module nor a secondary lattice module.

Proposition 7.
Let be a -lattice and let be a nonzero -lattice module. If for each compact ∈ , 1 = 1 or 1 = 0 , then is a second -lattice module.

Proposition 8. If
is a second -lattice module, then ( ) = (0 : 1 ) = is a prime element of . In this case, is called -second lattice module.
Proof. Suppose that is a second -lattice module. Clearly, ( ) = is a proper element of . Let ≤ and assume that ; that is, 1 ̸ = 0 . But is a second -lattice module; then 1 = 1 . Since 1 = 1 and 1 = 0 , then 1 = 0 , which implies that ≤ . is pure, = 1 ⋀ . As is a secondary lattice module, then either 1 = 1 or there exists a positive integer such that 1 = 0 . This implies that either = or = 1 ⋀ = 0 . Therefore, we have    Definition 15. An -module is called a multiplication lattice module if for every element ∈ , there exists an element ∈ , such that = 1 .
Definition 16. A element of an -module is called prime element if ̸ = 1 and whenever ∈ and ∈ with ≤ , then ≤ or ≤ ( : 1 ).

Definition 17. A element
of an -module is called semiprime element if ̸ = 1 and whenever ∈ and ∈ with 2 ≤ , then ≤ .
Remark 18. Let be a proper element of an -module .
Then is a semiprime element if and only if whenever ∈ , ∈ and is a positive integer with ≤ , then ≤ .
We know that a prime element is semiprime, but the converse is not true in general. The following proposition shows that the converse is true when the module is secondary and multiplication. Proof. ⇒: Suppose that is a semiprime element of and let ≤ , where ∈ , ∈ . Since is a secondary lattice module, then either 1 = 0 for some positive integer or 1 = 1 . Therefore, is a prime element of . ⇐: It is obvious.
Definition 20. Let be an -lattice module and let be a proper element of . is called a primary element of , if whenever ∈ , ∈ such that ≤ , then ≤ or ≤ √( : 1 ). Particularly, if is nonzero and 0 is primary, then is said to be primary lattice module. Proof. Let be a multiplication secondary module and = 0 for some ∈ , ∈ . Now, we assume that √ ( ). Since is a secondary module, then we have 1 = 1 . Because is a multiplication, then we have = . Consequently, we obtain = 0 .
Proposition 23. Every multiplication second lattice module is a simple lattice module.
Proof. Let be a multiplication and second module. Since is a multiplication, for every ∈ , there exists ∈ such that = 1 . Then we obtain 1 = 1 or 1 = 0 , since is second. Thus, we have = 1 or = 0 for every ∈ ; that is, is simple.
Definition 24. Let be a domain and let be a nonzerolattice module. If 1 = 1 for every 0 ̸ = ∈ , then is said to be divisible.

Definition 25. A nonzero -lattice module
is said to be torsion if there exists 0 ̸ = ∈ such that 1 = 0 .

Proposition 26. Let be a domain. Let be a secondarylattice module. Then either is a divisible module or is a torsion module.
Proof. Suppose that is a secondary module over a domain . If is not divisible, then there exists 0 ̸ = ∈ such that 1 ̸ = 1 . Since is a secondary lattice module, then there exists a positive integer such that 1 = 0 . Since 0 ̸ = and is a domain, then we have ̸ = 0 . Consequently, there exists 0 ̸ = = ∈ such that 1 = 0 . Therefore, is a torsion lattice module.