# ϕ-Prime and ϕ-Primary Elements in Multiplicative Lattices

Authors: C. S. Manjarekar; A. V. Bingi
Journal: Algebra
 (2014)

---

## Abstract

We investigateϕ-prime and ϕ-primary elements in a compactly generated multiplicative lattice L. By a counterexample, it is shown that a ϕ-primary element in L need not be primary. Some characterizations of ϕ-primary and ϕ-prime elements in L are obtained. Finally, some results for almost prime and almost primary elements in L with characterizations are obtained.


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## Body

## 1. Introduction

A multiplicative latticeL 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
⩽
∨
X implies the existence of a finite number of elements a
1
,
a
2
,
…
,
a
n in X such that a
⩽
a
1
∨
a
2
∨
⋯
∨
a
n. The set of compact elements of L will be denoted by L
*. A multiplicative lattice is said to be compactly generated if every element of it is a join of compact elements. Throughout this paper L denotes a compactly generated multiplicative lattice with 1 compact in which every finite product of compact elements is compact.An elementa
∈
L is said to be proper if a
<
1. A proper element p
∈
L is called a prime element if a
b
⩽
p implies a
⩽
p or b
⩽
p, where a
,
b
∈
L, and is called a primary element if a
b
⩽
p implies a
⩽
p or b
n
⩽
p for some n
∈
Z
+, where a
,
b
∈
L
*. A proper element p
∈
L is said to be weakly prime if 0
≠
a
b
⩽
p implies either a
⩽
p or b
⩽
p, where a
,
b
∈
L, and is called weakly primary if 0
≠
a
b
⩽
p implies a
⩽
p or b
n
⩽
p for some n
∈
Z
+, where a
,
b
∈
L
*. For a
,
b
∈
L, (
a
:
b
)
=
∨
{
x
∈
L
∣
x
b
⩽
a
}. The radical of a
∈
L is denoted by a and is defined as ∨
{
x
∈
L
*
∣
x
n
⩽
a
,
for
some
n
∈
Z
+
}
=
∧
{
p
∈
L
∣
a
⩽
p
and
p
is
a
prime
element
}. An element a
∈
L is called semiprimary if a is a prime element and is called semiprime if a
=
a. An element a
∈
L is called join irreducible if a
=
a
1
∨
a
2 implies a
=
a
1 or a
=
a
2. A proper element m
∈
L is said to be a maximal element if m

a for any other proper element a
∈
L. An element a
∈
L is said to be nilpotent if a
n
=
0 for some n
∈
Z
+. An element a
∈
L is called a zero divisor if a
b
=
0 for some 0
≠
b
∈
L and is called an idempotent if a
=
a
2. A multiplicative lattice is said to be a domain if it is without zero divisors and is said to be quasi-local if it contains a unique maximal element. A quasi-local multiplicative lattice L with maximal element m is denoted by (
L
,
m
). An element e
∈
L is called meet principal if a
∧
b
e
=
(
(
a
:
e
)
∧
b
)
e for all a
,
b
∈
L. An element e
∈
L is called join principal if (
a
e
∨
b
)
:
e
=
(
b
:
e
)
∨
a for all a
,
b
∈
L. An element e
∈
L is called weak meet principal if a
∧
e
=
(
a
:
e
)
e for all a
∈
L. An element e
∈
L is called weak join principal if (
a
e
:
e
)
=
(
0
:
e
)
∨
a for all a
∈
L. An element e
∈
L is called principal if e is both meet principal and join principal. An element e
∈
L is called weak principal if e is both weak meet principal and weak join principal. A multiplicative lattice is a Noether lattice if it is modular, principally generated (every element is a join of principal elements) and satisfies the ascending chain condition. A Noether lattice L is local if it contains precisely one maximal prime. In a Noether lattice L, an element a
∈
L is said to satisfy restricted cancellation law if, for any b
,
c
∈
L, a
b
=
a
c
≠
0 implies b
=
c [1]. The reader is referred to [2] for general background and terminology.


## 2.ϕ-Prime and ϕ-Primary Elements in L

The study ofϕ-prime and ϕ-primary ideals for commutative rings is carried out by Anderson and Bataineh [3], Bataineh and Kuhail [4], and Yousefian Darani [5]. We extend these concepts to compactly generated multiplicative lattices. We introduce the notion of ϕ-prime, ϕ-primary, and ω-primary elements in L.Definition 1.
Letϕ
:
L
→
L be a function. A proper element p
∈
L is said to be ϕ-prime if, for a
,
b
∈
L, a
b
⩽
p and a
b

ϕ
(
p
) implies either a
⩽
p or b
⩽
p.Definition 2.
Letϕ
:
L
→
L be a function. A proper element p
∈
L is said to be ϕ-primary if, for a
,
b
∈
L, a
b
⩽
p and a
b

ϕ
(
p
) implies either a
⩽
p or b
⩽
p.Definition 3.
Letϕ
:
L
→
L be a function. A proper element p
∈
L is said to be ϕ
ω-primary (or briefly ω-primary) if, for a
,
b
∈
L, a
b
⩽
p and a
b

⋀
n
=
1
∞
p
n
=
ϕ
ω
(
p
) implies either a
⩽
p or b
⩽
p.Similarly,ϕ
α-primary elements in L are defined by following settings in the above definition for ϕ-primary element. For any proper element p
∈
L,(i)
ϕ
0
(
p
)
=
0 and then p is called weakly primary element;
(ii)
ϕ
2
(
p
)
=
p
2 and then p is called almost primary element;
(iii)
ϕ
n
(
p
)
=
p
n
(
n
⩾
2
) and then p is called n-almost primary element.Thus an almost primary element inL is a particular case of a ϕ
α-primary element in L. Similarly an almost prime element in L is a particular case of a ϕ
α-prime element in L. Therefore their definitions are as follows.Definition 4.
A proper elementp
∈
L is said to be almost prime if, for a
,
b
∈
L, a
b
⩽
p and a
b

p
2 implies either a
⩽
p or b
⩽
p.Definition 5.
A proper elementp
∈
L is said to be almost primary if, for a
,
b
∈
L, a
b
⩽
p and a
b

p
2 implies either a
⩽
p or b
⩽
p.The reader can verify the following statements.(1)
ϕ
(
p
)
⩽
p for all p
∈
L.
(2)
Everyϕ-prime element in L is ϕ-primary.(Its converse holds for semiprime elements.)(3)
Every prime element inL is ϕ-prime.
(4)
Every prime element inL is ϕ-primary.
(5)
Every primary element inL is ϕ-primary.The following example (takeϕ
(
p
)
=
p
2 for convenience) shows that①
aϕ-primary element in L need not be ϕ-prime;
②
aϕ-primary element in L need not be prime.Example 6.
Consider the latticeL of ideals of the ring R
=
〈
Z
24
,
+
,
·
〉. Then the only ideals of R are the principal ideals (0), (2), (3), (4), (6), (8), (12), (1). Clearly L
=
{
(
0
)
,
(
2
)
,
(
3
)
,
(
4
)
,
(
6
)
,
(
8
)
,
(
12
)
,
(
1
)
} is a compactly generated multiplicative lattice. Its lattice structure is as shown in Figure 1.
From the multiplication table (see Figure1), we see that element c
∈
L is almost primary, while c is not almost prime because (
2
)
(
6
)
⩽
(
4
)
=
c but (
2
)

(
4
)
=
c and (
6
)

(
4
)
=
c. Also, c is not prime.Figure 1Theorem 7.
If a proper elementq
∈
L is ϕ-primary such that ϕ
(
q
)
=
ϕ
(
q
), then q is a ϕ-prime element in L.Proof.
Letp
=
q. Assume that a
b
⩽
p and a
b

ϕ
(
p
) but a

p for some a
,
b
∈
L. Then there exist n
∈
Z
+ such that (
a
b
)
n
⩽
q. If (
a
b
)
n
⩽
ϕ
(
q
), then by hypothesis a
b
⩽
ϕ
(
q
)
=
ϕ
(
p
), a contradiction. So (
a
b
)
n

ϕ
(
q
). Since q is ϕ-primary and a
n

q for all n
∈
Z
+, we have b
n
⩽
q and hence b
⩽
q
=
q. This shows that q is a ϕ-prime element in L.Definition 8.
Given two functionsγ
1
,
γ
2
:
L
→
L, we define γ
1
⩽
γ
2 if γ
1
(
p
)
⩽
γ
2
(
p
) for each p
∈
L.Theorem 9.
Letγ
1
,
γ
2
:
L
→
L be functions such that γ
1
⩽
γ
2. Then every proper γ
1-primary element in L is γ
2-primary.Proof.
Let a proper elementp
∈
L be γ
1-primary. Suppose a
b
⩽
p, a
b

γ
2
(
p
) for some a
,
b
∈
L. Then as γ
1
⩽
γ
2 we have a
b

γ
1
(
p
). Since p is γ
1-primary, it follows that either a
⩽
p or b
⩽
p and hence p is γ
2-primary.Theorem 10.
Letp
∈
L be a proper element. Then p is primary implies p is weakly primary implies p is ω-primary implies p is n-almost primary (
n
⩾
2
) implies p is almost primary.Proof.
Obviouslyp is primary implies p is weakly primary.
Assume thatp is weakly primary but not ω-primary. Then there exist a
,
b
∈
L such that a
b
⩽
p, a
b

⋀
n
=
1
∞
p
n, a

p, and b

p. As L is compactly generated, there exist x
,
y
∈
L
* such that x
⩽
a, y
⩽
b, x

p, andy

p. Let x
′
⩽
a and y
′
⩽
b be any two compact elements of L. Then (
x
∨
x
′
)
,
(
y
∨
y
′
)
∈
L
* such that (
x
∨
x
′
)
(
y
∨
y
′
)
⩽
a
b
⩽
p, (
x
∨
x
′
)

p, and(
y
∨
y
′
)

p. Since p is weakly primary, it follows that (
x
∨
x
′
)
(
y
∨
y
′
)
=
0. So x
′
y
′
=
0, which implies a
b
=
0
⩽
⋀
n
=
1
∞
p
n, a contradiction to a
b

⋀
n
=
1
∞
p
n. Hence p is ω-primary.
Next we show thatp is ω-primary implies p is n-almost primary (
n
⩾
2
). Assume p is ω-primary and n
⩾
2. Let a
b
⩽
p, a
b

p
n for some a
,
b
∈
L. Then a
b
⩽
p, a
b

⋀
n
=
1
∞
p
n. Since p is ω-primary, it follows that either a
⩽
p or b
⩽
p. Hence, p is n-almost primary (
n
⩾
2
).
The last implication is obvious (fromn
=
2).From this theorem we get the following characterization of aω-primary element in L.Corollary 11.
Letp
∈
L be a proper element. Then p is ω-primary if and only if p is n-almost primary for every n
⩾
2.Proof.
Supposep
∈
L is n-almost primary for every n
⩾
2. Let a
b
⩽
p and a
b

⋀
n
=
1
∞
p
n for some a
,
b
∈
L. Then a
b
⩽
p and a
b

p
n for some n
⩾
2. Since p is n-almost primary, we have a
⩽
p or b
⩽
p. Hence p is ω-primary. The converse follows from Theorem 10.Clearly every primary element inL is ϕ-primary. But the converse is not true as shown in the following example by taking ϕ
(
p
)
=
p
2.Example 12.
Consider the latticeL of ideals of the ring R
=
〈
Z
30
,
+
,
·
〉. Then the only ideals of R are the principal ideals (0), (2), (3), (5), (6), (10), (15), (1). Clearly L
=
{
(
0
)
,
(
2
)
,
(
3
)
,
(
5
)
,
(
6
)
,
(
10
)
,
(
15
)
,
(
1
)
} is a compactly generated multiplicative lattice. Here the element (
6
)
∈
L is almost primary but not primary.In the following successive three theorems, we show conditions under which aϕ-primary element in L is primary.Now we have a characterization of aϕ-primary element in L.Theorem 13.
LetL be a Noether lattice. Let 0
≠
a
∈
L be a non-nilpotent proper element satisfying the restricted cancellation law. Then a is ϕ-primary for some ϕ
⩽
ϕ
2 if and only if a is primary.Proof.
Supposea
∈
L is a primary element. Then obviously a is ϕ-primary for every ϕ and hence for ϕ
⩽
ϕ
2. Conversely, let a be ϕ-primary for some ϕ
⩽
ϕ
2. Then, by Theorem 9, a is ϕ
2-primary (almost primary). Let x
y
⩽
a for some x
,
y
∈
L
*. If x
y

a
2, then as a is ϕ
2-primary we have x
⩽
a or y
⩽
a. If x
y
⩽
a
2, consider (
x
∨
a
)
y
=
x
y
∨
a
y
⩽
a. If (
x
∨
a
)
y

a
2, then as a is ϕ
2-primary we have (
x
∨
a
)
⩽
a or y
m
⩽
a for some m
∈
Z
+ and hence x
⩽
a or y
⩽
a. So assume (
x
∨
a
)
y
⩽
a
2. Then x
y
⩽
a
2 and a
y
⩽
a
2
≠
0. By Lemma 1.11 of [1] we get y
⩽
a which shows that a is primary.Definition 14.
A proper elementp
∈
L is said to be 2-potent prime if, for a
,
b
∈
L, a
b
⩽
p
2 implies either a
⩽
p or b
⩽
p.Definition 15.
A proper elementp
∈
L is said to be 2-potent primary if, for a
,
b
∈
L, a
b
⩽
p
2 implies either a
⩽
p or b
⩽
p.Theorem 16.
Let a proper elementq
∈
L be 2-potent primary. If q is ϕ-primary for some ϕ
⩽
ϕ
2, then q is primary.Proof.
Clearly by Theorem9, q
∈
L is ϕ
2-primary (almost primary). Let x
y
⩽
q for some x
,
y
∈
L
*. If x
y

q
2, then as q is ϕ
2-primary we have x
⩽
q or y
⩽
q. If x
y
⩽
q
2, then as q is 2-potent primary we have x
⩽
q or y
⩽
q. Hence q is primary.Theorem 17.
Let a proper elementq
∈
L be ϕ-primary. If q
2

ϕ
(
q
), then q is primary.Proof.
Leta
b
⩽
q for some a
,
b
∈
L
*. If a
b

ϕ
(
q
), then as q is ϕ-primary we have a
⩽
q or b
⩽
q. So assume a
b
⩽
ϕ
(
q
). First suppose a
q

ϕ
(
q
). Then a
d

ϕ
(
q
) for some d
⩽
q in L. Also a
(
b
∨
d
)
=
a
b
∨
a
d
⩽
q and a
(
b
∨
d
)

ϕ
(
q
). As q is ϕ-primary, either a
⩽
q or (
b
∨
d
)
k
⩽
q for some k
∈
Z
+. Hence a
⩽
q or b
k
⩽
q for some k
∈
Z
+. Similarly if b
q

ϕ
(
q
) we can show that either a
⩽
q or b
k
⩽
q for some k
∈
Z
+. So we can assume that a
q
⩽
ϕ
(
q
) and b
q
⩽
ϕ
(
q
). Since q
2

ϕ
(
q
), there exist r
,
s
⩽
q in L such that r
s

ϕ
(
q
). Then (
a
∨
r
)
(
b
∨
s
)
⩽
q, but (
a
∨
r
)
(
b
∨
s
)

ϕ
(
q
). As q is ϕ-primary, we have either (
a
∨
r
)
⩽
q or (
b
∨
s
)
t
⩽
q for some t
∈
Z
+. Hence a
⩽
q or b
t
⩽
q for some t
∈
Z
+. Thus q is primary.From the above theorem it follows that①
if a proper elementq
∈
L is ϕ-primary but not primary, then q
2
⩽
ϕ
(
q
);
②
aϕ-primary element q
<
1 in L with q
2

ϕ
(
q
) is primary.Corollary 18.
If a proper elementq
∈
L is ϕ-primary but not primary, then q
=
ϕ
(
q
).Proof.
From Theorem17 we have q
2
⩽
ϕ
(
q
). So q
⩽
ϕ
(
q
) which gives q
⩽
ϕ
(
q
)
=
ϕ
(
q
). Since ϕ
(
q
)
⩽
q, we have ϕ
(
q
)
⩽
q. Hence q
=
ϕ
(
q
).Corollary 19.
If a proper elementq
∈
L is ϕ-primary where ϕ
⩽
ϕ
3, then q is ω-primary.Proof.
Ifq is primary, then by Theorem 10, q is ω-primary. So assume that q is not primary. Then by Theorem 17 and hypothesis we get q
2
⩽
ϕ
(
q
)
⩽
q
3. Hence ϕ
(
q
)
=
q
n for every n
⩾
2. Consequently q is n-almost primary for every n
⩾
2 and hence it is ω-primary by Corollary 11.Corollary 20.
If a proper elementq
∈
L is ϕ
0-primary but not primary, then q
2
=
0.Proof.
The proof is obvious.Now we obtain the characterization of aϕ-primary element in L.Theorem 21.
Letq
∈
L be a proper element and let ϕ
:
L
→
L be a function. Then the following statements are equivalent:①
q is ϕ-primary;
②
for everya
∈
L such that a

q, either (
q
:
a
)
=
q or (
q
:
a
)
=
(
ϕ
(
q
)
:
a
);
③
for any two elementsr
,
s
∈
L
*, r
s
⩽
q and r
s

ϕ
(
q
) implies either s
⩽
q or r
⩽
q.Proof.
①⇒②. Suppose ① holds. Let h
∈
L be compact such that h
⩽
(
q
:
a
) and a

q. Then a
h
⩽
q. If a
h
⩽
ϕ
(
q
), then h
⩽
(
ϕ
(
q
)
:
a
). If a
h

ϕ
(
q
), then since q is ϕ-primary and a

q it follows that h
⩽
q. Hence by Lemma 1 of [6] either (
q
:
a
)
⩽
(
ϕ
(
q
)
:
a
) or (
q
:
a
)
⩽
q. Consequently either (
q
:
a
)
=
(
ϕ
(
q
)
:
a
) or (
q
:
a
)
=
q.
②⇒③. Suppose ② holds. Let r
s
⩽
q, r
s

ϕ
(
q
), and r

q for some r
,
s
∈
L
*. Then by ② we have either (
q
:
r
)
=
(
ϕ
(
q
)
:
r
) or (
q
:
r
)
=
q. If (
q
:
r
)
=
(
ϕ
(
q
)
:
r
), then as s
⩽
(
q
:
r
) it follows that s
⩽
(
ϕ
(
q
)
:
r
) which contradicts r
s

ϕ
(
q
). If (
q
:
r
)
=
q, then s
⩽
(
q
:
r
) gives s
⩽
q.
③⇒①. Suppose ③ holds. Let a
b
⩽
q, a
b

ϕ
(
q
), and a

q for some a
,
b
∈
L. Then as L is compactly generated, there exist x
,
x
′
,
y
′
∈
L
* such that x
⩽
a, x
′
⩽
a, and y
′
⩽
b but x

q, x
′
y
′

ϕ
(
q
). Let y
⩽
b be any compact element of L. Then (
x
∨
x
′
)
,
(
y
∨
y
′
)
∈
L
* such that (
x
∨
x
′
)
(
y
∨
y
′
)
⩽
a
b
⩽
q, (
x
∨
x
′
)
(
y
∨
y
′
)
⩽
a
b

ϕ
(
q
), and (
x
∨
x
′
)

q. So by ③ (
y
∨
y
′
)
⩽
q which implies b
⩽
q. Therefore, q is ϕ-primary.Likeω-primary element in L we introduce the notion of ω-prime element in L.Definition 22.
Letϕ
:
L
→
L be a function. A proper element p
∈
L is said to be ϕ
ω-prime (or briefly ω-prime) if, for a
,
b
∈
L, a
b
⩽
p and a
b

⋀
n
=
1
∞
p
n
=
ϕ
ω
(
p
) implies either a
⩽
p or b
⩽
p.Similarlyϕ
α-prime elements in L are defined by following settings in the definition for ϕ-prime element. For any proper element p
∈
L,(i)
ϕ
0
(
p
)
=
0 and then p is called weakly prime element;
(ii)
ϕ
2
(
p
)
=
p
2 and then p is called almost prime element;
(iii)
ϕ
n
(
p
)
=
p
n
(
n
⩾
2
) and then p is called n-almost prime element.The analogous theorems and corollaries (obtained forϕ-primary elements in L) for ϕ-prime elements in L are as follows whose proofs being on similar arguments are omitted.Theorem 23.
Letγ
1
,
γ
2
:
L
→
L be functions such that γ
1
⩽
γ
2. Then every proper γ
1-prime element in L is γ
2-prime.Theorem 24.
Letp
∈
L be a proper element. Then p is prime implies p is weakly prime implies p is ω-prime implies p is n-almost prime (
n
⩾
2
) implies p is almost prime.Now we have the characterization of anω-prime element in L.Corollary 25.
Letp
∈
L be a proper element. Then p is ω-prime if and only if p is n-almost prime for every n
⩾
2.Clearly every prime element inL is ϕ-prime. But the converse is not true as shown in the following example by taking ϕ
(
p
)
=
p
2.Example 26.
ConsiderL as in Example 6. Here the element (
0
)
∈
L is weakly prime and hence almost prime, while (0) is not prime since (
2
)
(
12
)
⩽
(
0
), but neither (
2
)
⩽
(
0
) nor (
12
)
⩽
(
0
).In the following successive three theorems we show conditions under which aϕ-prime element in L is prime.The next theorem gives a characterization of aϕ-prime element in L.Theorem 27.
LetL be a Noether lattice. Let 0
≠
a
∈
L be a non-nilpotent proper element satisfying the restricted cancellation law. Then a is ϕ-prime for some ϕ
⩽
ϕ
2 if and only if a is prime.Theorem 28.
Let a proper elementq
∈
L be 2-potent prime. If q is ϕ-prime for some ϕ
⩽
ϕ
2, then q is prime.Theorem 29.
Let a proper elementq
∈
L be ϕ-prime. If q
2

ϕ
(
q
), then q is prime.From the above theorem it follows that①
if a proper elementq
∈
L is ϕ-prime but not prime, then q
2
⩽
ϕ
(
q
);
②
aϕ-prime element q
<
1 in L with q
2

ϕ
(
q
) is prime.Corollary 30.
If a proper elementq
∈
L is ϕ-prime but not prime, then q
=
ϕ
(
q
).Corollary 31.
If a proper elementq
∈
L is ϕ-prime where ϕ
⩽
ϕ
3, then q is ω-prime.Corollary 32.
If a proper elementq
∈
L is ϕ
0-prime but not prime, then q
2
=
0.Theorem 33.
Letq
∈
L be a proper element and let ϕ
:
L
→
L be a function. Then the following statements are equivalent:①
q is ϕ-prime;
②
for everya
∈
L such that a

q, either (
q
:
a
)
=
q or (
q
:
a
)
=
(
ϕ
(
q
)
:
a
);
③
for any two elementsr
,
s
∈
L
*, r
s
⩽
q and r
s

ϕ
(
q
) implies either s
⩽
q or r
⩽
q.


## 3. Almost Prime and Almost Primary Elements inL

The study of almost prime and almost primary ideals for commutative rings is carried out by Bhatwadekar and Sharma [7] and Bataineh and Kuhail [4], respectively. From the previous section we know that almost prime and almost primary elements in L are particular cases of ϕ
α-prime and ϕ
α-primary elements in L, respectively. In this section, we obtain some results on an almost prime element and on an almost primary element in L with characterizations.The reader can verify the following statements.(1)
Every almost prime element inL is almost primary.(Its converse holds for semiprime elements.)(2)
Every prime element inL is almost prime.
(3)
Every idempotent element inL is almost prime.
(4)
Every prime element inL is almost primary.
(5)
Every idempotent element inL is almost primary.But the converse is not true as shown in the following example.Example 34.
Consider the latticeL of ideals of the ring R
=
〈
Z
8
,
+
,
·
〉. Then the only ideals of R are the principal ideals (
0
)
,
(
2
)
,
(
4
)
,
(
1
). Clearly L
=
{
(
0
)
,
(
2
)
,
(
4
)
,
(
1
)
} is a compactly generated multiplicative lattice. Here the element (
4
)
∈
L is almost primary but not idempotent. (6)
Every primary element inL is almost primary.But the converse is not true, which is clear from Example12.In the following successive three theorems we show conditions under which an almost primary element inL is primary.Theorem 35.
Let a proper elementp
∈
L be 2-potent primary. Then p being almost primary implies that p is primary.Proof.
The proof is obvious.Theorem 36.
Let a proper elementp
∈
L be 2-potent prime. Then①
p being almost prime implies that p is prime;
②
p being almost primary implies that p is primary.Proof.
The proof is obvious.From the following examples it is clear that①
an almost primary element inL need not be 2-potent prime;
②  a
2-potent prime element inL which is almost primary need not be prime.Example 37.
ConsiderL as in Example 12. Here the element (
6
)
∈
L is almost primary but not 2-potent prime.Example 38.
ConsiderL as in Example 34. Here the element (
4
)
∈
L is 2-potent prime, almost primary but not prime.Theorem 39.
LetL be a local Noetherian domain. If a proper element p
∈
L is n-almost primary (
n
⩾
2
), then p is primary.Proof.
Leta
b
⩽
p for some a
,
b
∈
L
*. If a
b

p
n for all n
⩾
2, then as p
∈
L is n-almost primary we have a
⩽
p or b
⩽
p. If a
b
⩽
p
n for all n
⩾
1, then as L is a local Noetherian domain, from Corollary 3.3 of [8], it follows that a
b
⩽
⋀
n
=
1
∞
p
n
=
{
0
}. Thus a
b
=
0. Since L is domain, we have either a
=
0 or b
=
0 which implies a
⩽
p or b
⩽
p and hence p is primary.Theorem 40.
Let(
L
,
m
) be a quasi-local Noether lattice. If a proper element p
∈
L is such that p
2
=
m
2
⩽
p
⩽
m, then p is almost primary.Proof.
Letx
y
⩽
p and x
y

p
2 for some x
,
y
∈
L. If x

m, then x
=
1. So x
y
⩽
p gives y
⩽
p. Similarly y

m gives x
⩽
p. Now if x
⩽
m, then x
2
⩽
m
2
=
p
2
⩽
p and hence x
⩽
p. Similarly y
⩽
m gives y
⩽
p. Hence in any case p is almost primary.The next theorem gives the characterization of an almost primary element inL.Theorem 41.
Let a proper elementq
∈
L be join irreducible. Then the following statements are equivalent:①
q is almost primary;
②
for everya
∈
L such that a

q, (
q
:
a
)
=
q
∨
(
q
2
:
a
);
③
for everya
∈
L such that a

q, either (
q
:
a
)
=
q or (
q
:
a
)
=
(
q
2
:
a
).Proof.
①⇒②. Suppose ① holds. Let h
∈
L be compact such that h
⩽
(
q
:
a
) and a

q. Then a
h
⩽
q. If a
h
⩽
q
2, then h
⩽
(
q
2
:
a
). If a
h

q
2, then since q is almost primary and a

q it follows that h
⩽
q. Hence by Lemma 1 of [6] either (
q
:
a
)
⩽
(
q
2
:
a
)
⩽
q
∨
(
q
2
:
a
) or (
q
:
a
)
⩽
q
⩽
q
∨
(
q
2
:
a
). But as q
∨
(
q
2
:
a
)
⩽
(
q
:
a
), we have (
q
:
a
)
=
q
∨
(
q
2
:
a
).
②⇒③. Suppose ② holds. Let a

q for some a
∈
L. Then by ② (
q
:
a
)
=
q
∨
(
q
2
:
a
). Now as q is join irreducible, we have either (
q
:
a
)
=
q or (
q
:
a
)
=
(
q
2
:
a
).
③⇒①. Suppose ③ holds. Let r
s
⩽
q, r
s

q
2, and r

q for some r
,
s
∈
L. Then by ③ we have either (
q
:
r
)
=
(
q
2
:
r
) or (
q
:
r
)
=
q. If (
q
:
r
)
=
(
q
2
:
r
), then as s
⩽
(
q
:
r
) it follows that s
⩽
(
q
2
:
r
) which contradicts r
s

q
2. If (
q
:
r
)
=
q, then s
⩽
(
q
:
r
) gives s
⩽
q. Therefore q is almost primary.In view of Proposition 2 of [9] and the above Theorems 21 and 41 we have the following result.Corollary 42.
Let(
L
,
m
) be a quasi-local Noether lattice. Let a proper element q
∈
L be weak meet principal and let f
⩽
q be a nonzero weak join principal element in L. Then the following statements are equivalent:①
q is almost primary;
②
for everya
∈
L such that a

q, (
q
:
a
)
=
q
∨
(
q
2
:
a
);
③
for everya
∈
L such that a

q, either (
q
:
a
)
=
q or (
q
:
a
)
=
(
q
2
:
a
);
④
for any two elementsr
,
s
∈
L
*, r
s
⩽
q and r
s

q
2 implies either s
⩽
q or r
⩽
q.Proof.
The proof is obvious.The following theorem gives the similar characterization of an almost primary element inL.Theorem 43.
A proper elementq
∈
L is almost primary if and only if for every a
∈
L such that a

q either (
q
:
a
)
⩽
q or (
q
:
a
)
=
(
q
2
:
a
).Proof.
Assume that the proper elementq
∈
L is almost primary. Let h
∈
L be compact such that h
⩽
(
q
:
a
) and a

q. Then a
h
⩽
q. If a
h
⩽
q
2, then h
⩽
(
q
2
:
a
). If a
h

q
2, then since q is almost primary and a

q it follows that h
⩽
q. Hence by Lemma 1 of [6] either (
q
:
a
)
⩽
(
q
2
:
a
) or (
q
:
a
)
⩽
q. But as (
q
2
:
a
)
⩽
(
q
:
a
), we have (
q
:
a
)
⩽
q or (
q
:
a
)
=
(
q
2
:
a
). Conversely assume that for every a
∈
L such that a

q either (
q
:
a
)
⩽
q or (
q
:
a
)
=
(
q
2
:
a
). Let r
s
⩽
q, r
s

q
2, and r

q for some r
,
s
∈
L. Then either (
q
:
r
)
=
(
q
2
:
r
) or (
q
:
r
)
⩽
q. If (
q
:
r
)
=
(
q
2
:
r
), then as s
⩽
(
q
:
r
) it follows that s
⩽
(
q
2
:
r
) which contradicts r
s

q
2. If (
q
:
r
)
⩽
q, then s
⩽
(
q
:
r
) gives s
⩽
q. Therefore q is almost primary.The analogous theorems and corollaries (obtained for almost primary elements inL) for almost prime elements in L are as follows whose proofs being on similar arguments are omitted.Theorem 44.
LetL be a local Noetherian domain. If a proper element p
∈
L is n-almost prime (
n
⩾
2
), then p is prime.Theorem 45.
Let(
L
,
m
) be a quasi-local Noether lattice. If a proper element p
∈
L is such that p
2
=
m
2
⩽
p
⩽
m, then p is almost prime.The next theorem gives the characterization of an almost prime element inL.Theorem 46.
Let a proper elementq
∈
L be join irreducible. Then the following statements are equivalent:①
q is almost prime;
②
for everya
∈
L such that a

q, (
q
:
a
)
=
q
∨
(
q
2
:
a
);
③
for everya
∈
L such that a

q, either (
q
:
a
)
=
q or (
q
:
a
)
=
(
q
2
:
a
).Corollary 47.
Let(
L
,
m
) be a quasi-local Noether lattice. Let a proper element q
∈
L be weak meet principal and let f
⩽
q be a nonzero weak join principal element in L. Then the following statements are equivalent:①
q is almost prime;
②
for everya
∈
L such that a

q, (
q
:
a
)
=
q
∨
(
q
2
:
a
);
③
for everya
∈
L such that a

q, either (
q
:
a
)
=
q or (
q
:
a
)
=
(
q
2
:
a
);
④
for any two elementsr
,
s
∈
L
*, r
s
⩽
q and r
s

q
2 implies either s
⩽
q or r
⩽
q.



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