Hindawi Publishing Corporation
Algebra
Volume 2015, Article ID 183930, 6 pages
http://dx.doi.org/10.1155/2015/183930
Research Article
On 𝜙-Absorbing Primary Elements in Lattice Modules
Sachin Ballal and Vilas Kharat
Department of Mathematics, Savitribai Phule Pune University, Pune 411 007, India
Correspondence should be addressed to Sachin Ballal; [email protected]
Received 18 December 2014; Accepted 31 March 2015
Academic Editor: Andrei V. Kelarev
Copyright © 2015 S. Ballal and V. Kharat. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Let 𝐿 be a 𝐶-lattice and let 𝑀 be a lattice module over 𝐿. Let 𝜙 : 𝑀 → 𝑀 be a function. A proper element 𝑃 ∈ 𝑀 is said
to be 𝜙-absorbing primary if, for 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ∈ 𝐿 and 𝑁 ∈ 𝑀, 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃 and 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰ 𝜙(𝑃) together imply
√𝑃, for some 𝑖 ∈ {1, 2, . . . , 𝑛}. We study some basic properties of 𝜙-absorbing
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤ (𝑃 : 1𝑀 ) or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑀
primary elements. Also, various generalizations of prime and primary elements in multiplicative lattices and lattice modules as
𝜙-absorbing elements and 𝜙-absorbing primary elements are unified.
1. Introduction
A lattice 𝐿 is said to be complete, if, for any subset 𝑆 of 𝐿,
we have ∨𝑆, ∧𝑆 ∈ 𝐿. Since every complete lattice is bounded,
1𝐿 (or 1) denotes the greatest element and 0𝐿 (or 0) denotes
the smallest element of 𝐿. A complete lattice 𝐿 is said to be a
multiplicative lattice, if there is a defined binary operation “⋅”
called multiplication on 𝐿 satisfying the following conditions:
(1) 𝑎 ⋅ 𝑏 = 𝑏 ⋅ 𝑎, for all 𝑎, 𝑏 ∈ 𝐿,
(2) 𝑎 ⋅ (𝑏 ⋅ 𝑐) = (𝑎 ⋅ 𝑏) ⋅ 𝑐, for all 𝑎, 𝑏, 𝑐 ∈ 𝐿,
(3) 𝑎 ⋅ (∨𝛼 𝑏𝛼 ) = ∨𝛼 (𝑎 ⋅ 𝑏𝛼 ), for all 𝑎, 𝑏𝛼 ∈ 𝐿,
(4) 𝑎 ⋅ 1 = 𝑎, for all 𝑎 ∈ 𝐿.
Henceforth, 𝑎 ⋅ 𝑏 will be simply denoted by 𝑎𝑏.
An element 𝑝 ≠ 1 of a multiplicative lattice 𝐿 is said to be
prime if 𝑎𝑏 ≤ 𝑝 implies either 𝑎 ≤ 𝑝 or 𝑏 ≤ 𝑝, for 𝑎, 𝑏 ∈ 𝐿.
Radical of an element 𝑎 ∈ 𝐿 is denoted by√𝑎 and is defined
as√𝑎 = ∨{𝑥 ∈ 𝐿 | 𝑥𝑛 ≤ 𝑎, for some 𝑛 ∈ Z+ }.
An element 𝑐 of a complete lattice 𝐿 is said to be compact
if 𝑐 ≤ ∨𝛼 𝑎𝛼 implies 𝑐 ≤ ∨𝑛𝑖=1 𝑎𝛼𝑖 , where 𝑛 ∈ Z+ . The set of all
compact elements of a lattice 𝐿 is denoted by 𝐿 ∗ . By a 𝐶-lattice
we mean a multiplicative lattice 𝐿 with a multiplicatively
closed set 𝑆 of compact elements which generates 𝐿 under
join.
A complete lattice 𝑀 is said to be a lattice module over
a multiplicative lattice 𝐿, if there is a multiplication between
elements of 𝑀 and 𝐿, denoted by 𝑎𝑁 for 𝑎 ∈ 𝐿 and 𝑁 ∈ 𝑀,
which satisfies the following properties:
(1) (𝑎𝑏)𝑁 = 𝑎(𝑏𝑁),
(2) (∨𝛼 𝑎𝛼 )(∨𝛽 𝑁𝛽 ) = ∨𝛼,𝛽 (𝑎𝛼 𝑁𝛽 ),
(3) 1𝐿 𝑁 = 𝑁,
(4) 0𝐿 𝑁 = 0𝑀, for all 𝑎, 𝑏, 𝑎𝛼 ∈ 𝐿 and for all 𝑁, 𝑁𝛽 ∈ 𝑀,
where 1𝑀 denotes the greatest element of 𝑀 and 0𝑀 denotes
the smallest element of 𝑀.
For 𝑁 ∈ 𝑀 and 𝑎 ∈ 𝐿, denote (𝑁 : 𝑎) = ∨{𝑋 ∈ 𝑀 :
𝑎𝑋 ≤ 𝑁}. For 𝑎, 𝑏 ∈ 𝐿, (𝑎 : 𝑏) = ∨{𝑥 ∈ 𝐿 : 𝑏𝑥 ≤ 𝑎} and
for 𝐴, 𝐵 ∈ 𝑀, (𝐴 : 𝐵) = ∨{𝑥 ∈ 𝐿 : 𝑥𝐵 ≤ 𝐴}. For 𝑁 ∈ 𝑀,
√𝑁 = ∨{𝑥 ∈ 𝐿 : 𝑥𝑛 1𝑀 ≤ 𝑁} for some positive integer 𝑛
and it is also denoted by√(𝑁 : 1𝑀). For 𝑁 ∈ 𝑀, we define
𝑀
√
𝑁 =√(𝑁 : 1𝑀)1𝑀. An element 𝑁 ∈ 𝑀 is said to be weak
join principal if it satisfies the following identity 𝑎∨(0𝑀 : 𝑁) =
(𝑎𝑁 : 𝑁) for all 𝑎 ∈ 𝐿.
A lattice module 𝑀 over a multiplicative lattice 𝐿 is called
a multiplication lattice module if for 𝑁 ∈ 𝑀 there exists an
element 𝑎 ∈ 𝐿 such that 𝑁 = 𝑎1𝑀.
An element 𝑁 ≠ 1𝑀 in 𝑀 is said to be prime if 𝑎𝑋 ≤ 𝑁
implies 𝑋 ≤ 𝑁 or 𝑎1𝑀 ≤ 𝑁, that is, 𝑎 ≤ (𝑁 : 1𝑀) for every
𝑎 ∈ 𝐿 and 𝑋 ∈ 𝑀.
An element 𝑁 ∈ 𝑀 is called compact if 𝑁 ≤ ∨𝛼 𝐴 𝛼 implies
𝑁 ≤ 𝐴 𝛼1 ∨ 𝐴 𝛼2 ∨ ⋅ ⋅ ⋅ ∨ 𝐴 𝛼𝑛 for some 𝛼1 , 𝛼2 , . . . , 𝛼𝑛 . If each
element of 𝑀 is a join of principal (compact) elements of
2
𝑀, then 𝑀 is called principally generated lattice (compactly
generated lattice).
Çallıalp et al. [1] studied the concepts of weakly prime and
almost prime elements in multiplicative lattices as extensions
of, respectively, weakly prime and almost prime ideals in
commutative rings. An element 𝑝 ≠ 1 in 𝐿 is said to be weakly
prime if 0 ≠ 𝑎𝑏 ≤ 𝑝 implies either 𝑎 ≤ 𝑝 or 𝑏 ≤ 𝑝 and almost
prime if 𝑎𝑏 ≤ 𝑝 and 𝑎𝑏 ≰ 𝑝2 implies either 𝑎 ≤ 𝑝 or 𝑏 ≤ 𝑝
for 𝑎, 𝑏 ∈ 𝐿. In [2], the authors generalized these concepts,
respectively, to weakly primary and almost primary elements
in multiplicative lattices. A proper element 𝑝 ∈ 𝐿 is said to be
weakly primary if 0 ≠ 𝑎𝑏 ≤ 𝑝 implies either 𝑎 ≤ 𝑝 or 𝑏 ≤√𝑝
and almost primary if 𝑎𝑏 ≤ 𝑝 and 𝑎𝑏 ≰ 𝑝2 implies either 𝑎 ≤ 𝑝
or 𝑏 ≤√𝑝 for 𝑎, 𝑏 ∈ 𝐿.
The concept of prime elements in multiplicative lattices
is further generalised to 2-absorbing and weakly 2-absorbing
elements in multiplicative lattices by Jayaram et al. [3]. An
element 𝑞 < 1 in 𝐿 is said to be 2-absorbing if 𝑎𝑏𝑐 ≤ 𝑞 implies
either 𝑎𝑏 ≤ 𝑞 or 𝑏𝑐 ≤ 𝑞 or 𝑎𝑐 ≤ 𝑞 and is said to be weakly
2-absorbing element if 0 ≠ 𝑎𝑏𝑐 ≤ 𝑞 implies either 𝑎𝑏 ≤ 𝑞 or
𝑏𝑐 ≤ 𝑞 or 𝑎𝑐 ≤ 𝑞, for 𝑎, 𝑏, 𝑐 ∈ 𝐿. Joshi and Ballal [4] defined
the concept of 𝑛-prime elements in multiplicative lattices as
a generalization of prime elements. An element 𝑝 < 1 of
a multiplicative lattice 𝐿 is said to be an 𝑛-prime if we can
express it as meet of at most 𝑛 primes, where 𝑛 is a positive
integer.
In [5], the authors defined the concepts of 𝑛-absorbing,
weakly 𝑛-absorbing, and 𝑛-almost 𝑛-absorbing elements in
multiplicative lattices as generalizations of, respectively, 2absorbing, weakly 2-absorbing, and almost prime elements,
where 𝑛 ≥ 2. An element 𝑝 < 1 of a multiplicative
lattice 𝐿 is called 𝑛-absorbing if 𝑥1 𝑥2 𝑥3 ⋅ ⋅ ⋅ 𝑥𝑛+1 ≤ 𝑝 implies
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛+1 ≤ 𝑝 and called weakly 𝑛-absorbing
if, for 𝑥1 , 𝑥2 , 𝑥3 , . . . , 𝑥𝑛+1 ∈ 𝐿, 0 ≠ 𝑥1 𝑥2 𝑥3 ⋅ ⋅ ⋅ 𝑥𝑛+1 ≤
𝑝 implies 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛+1 ≤ 𝑝, 𝑖 ∈ {1, 2, . . . , 𝑛},
and 𝑥1 , 𝑥2 , 𝑥3 , . . . , 𝑥𝑛+1 ∈ 𝐿. An element 𝑝 < 1 of
a multiplicative lattice 𝐿 is called 𝑛-almost 𝑛-absorbing if
𝑥1 𝑥2 𝑥3 ⋅ ⋅ ⋅ 𝑥𝑛+1 ≤ 𝑝 and 𝑥1 𝑥2 𝑥3 ⋅ ⋅ ⋅ 𝑥𝑛+1 ≰ 𝑝𝑛 together
imply 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛+1 ≤ 𝑝 for 𝑖 ∈ {1, 2, . . . , 𝑛},
𝑥1 , 𝑥2 , 𝑥3 , . . . , 𝑥𝑛+1 ∈ 𝐿.
Manjarekar and Bingi [6] unified the theory of generalizations of prime and primary elements in multiplicative lattice
as 𝜙-prime and 𝜙-primary elements. Let 𝜙 : 𝐿 → 𝐿 be a
function. A proper element 𝑝 ∈ 𝐿 is said to be 𝜙-prime if
𝑎𝑏 ≤ 𝑝 and 𝑎𝑏 ≰ 𝜙(𝑝) implies either 𝑎 ≤ 𝑝 or 𝑏 ≤ 𝑝 and it is
said to be 𝜙-primary if 𝑎𝑏 ≤ 𝑝 and 𝑎𝑏 ≰ 𝜙(𝑝) implies either
𝑎 ≤ 𝑝 or 𝑏 ≤√𝑝, for 𝑎, 𝑏 ∈ 𝐿.
As a generalization of primary ideals in commutative
rings, Badawi et al. [7] introduced the concept of 2-absorbing
primary ideals. In this paper, we extend the various generalizations of prime ideals and primary ideals in commutative
rings to lattice modules. Also, we unify various generalizations of prime and primary elements in multiplicative lattices
and lattice modules, respectively, as 𝜙-absorbing elements
and 𝜙-absorbing primary elements.
For basic concepts and terminologies of lattice modules,
one may refer to [8–11] and for multiplicative lattices, one may
refer to [12–14].
Algebra
2. 𝜙-Absorbing Primary Elements
We introduce the concepts of 𝜙-absorbing elements and
𝜙-absorbing primary elements in lattice modules which
generalizes, respectively, the concepts of prime and primary
elements in multiplicative lattices and lattice modules (see [1–
3, 5, 6]).
Essentially, we have the following definitions, where 𝑀 is
a lattice module over a multiplicative lattice 𝐿 and 𝑛 ≥ 1.
Definition 1. A proper element 𝑃 ∈ 𝑀 is said to be 𝑛absorbing if 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃 implies 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤ (𝑃 : 1𝑀)
or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃 for 𝑖 ∈ {1, 2, . . . , 𝑛}, where
𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ∈ 𝐿 and 𝑁 ∈ 𝑀.
Definition 2. Let 𝜙 : 𝑀 → 𝑀 be a function. A proper
element 𝑃 ∈ 𝑀 is said to be 𝜙-absorbing if 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃
and 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰ 𝜙(𝑃) together imply 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤ (𝑃 :
1𝑀) or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃 for 𝑖 ∈ {1, 2, . . . , 𝑛},
where 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ∈ 𝐿 and 𝑁 ∈ 𝑀.
Definition 3. A proper element 𝑃 ∈ 𝑀 is said to be 𝑛absorbing primary if 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃 implies 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤
𝑀
(𝑃 : 1𝑀) or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ √
𝑃 for 𝑖 ∈ {1, 2, . . . , 𝑛},
where 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ∈ 𝐿 and 𝑁 ∈ 𝑀.
Definition 4. Let 𝜙 : 𝑀 → 𝑀 be a function. A proper
element 𝑃 ∈ 𝑀 is said to be 𝜙-absorbing primary if
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃 and 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰ 𝜙(𝑃) together imply
𝑀
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤ (𝑃 : 1𝑀) or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ √
𝑃 for
𝑖 ∈ {1, 2, . . . , 𝑛}, where 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ∈ 𝐿 and 𝑁 ∈ 𝑀.
Let 𝑀 be a lattice module over a multiplicative lattice 𝐿.
For a map 𝜙𝛼 : 𝑀 → 𝑀, we have the following.
(1) 𝜙0 : 𝜙(𝑃) = 0 defines weakly 𝑛-absorbing primary
elements of 𝑀.
(2) 𝜙2 : 𝜙(𝑃) = (𝑃 : 1𝑀)𝑃 defines almost 𝑛-absorbing
primary elements of 𝑀.
(3) 𝜙𝑚+1 (𝑛 ≥ 1) : 𝜙(𝑃) = (𝑃 : 1𝑀)𝑚 𝑃 defines 𝑚-almost
𝑛-absorbing primary elements of 𝑀.
𝑚
(4) 𝜙𝜔 : 𝜙(𝑃) = ∧∞
𝑚=1 (𝑃 : 1𝑀 ) 𝑃 defines 𝜔-absorbing
primary elements of 𝑀.
Remark 5. It follows immediately from the definition that any
𝑛-absorbing primary element of 𝑀 is a 𝜙-absorbing primary.
However, the converse does not necessarily holds. In fact,
we have the following theorem in which the converse is true
under certain condition.
Theorem 6. Let 𝑀 be a lattice module over a 𝐶-lattice 𝐿. Then
every 𝜙-absorbing primary element 𝑃 ∈ 𝑀 with (𝑃 : 1𝑀)𝑛 𝑃 ≰
𝜙(𝑃) is 𝑛-absorbing primary.
Proof. Suppose that (𝑃 : 1𝑀)𝑛 𝑃 ≰ 𝜙(𝑃) and 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤
𝑃, for 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ∈ 𝐿, 𝑁 ∈ 𝑀. If 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰ 𝜙(𝑃), then
as 𝑃 is 𝜙-absorbing primary, we have 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤ (𝑃 : 1𝑀)
𝑀
or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ √
𝑃 for some 𝑖 ∈ {1, 2, . . . , 𝑛},
which concludes that 𝑃 is 𝑛-absorbing primary.
Algebra
3
Now, suppose that 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝜙(𝑃). We assume that
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−𝑘 (𝑃 : 1𝑀)𝑘 𝑁 ≤ 𝜙(𝑃), for all 𝑘 ∈ {1, 2, . . . , 𝑛 −
1}. If 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−𝑘 (𝑃 : 1𝑀)𝑘 𝑁 ≰ 𝜙(𝑃), then there exists
𝑎1 , 𝑎2 , . . . , 𝑎𝑘 ≤ (𝑃 : 1𝑀) such that 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−𝑘 𝑎1 𝑎2 ⋅ ⋅ ⋅ 𝑎𝑘 𝑁 ≰
𝜙(𝑃). Hence, 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−𝑘 (𝑥𝑛−𝑘+1 ∨ 𝑎1 )(𝑥𝑛−𝑘+2 ∨ 𝑎2 ) ⋅ ⋅ ⋅ (𝑥𝑛 ∨
𝑎𝑘 )𝑁 ≤ 𝑃 and 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−𝑘 (𝑥𝑛−𝑘+1 ∨ 𝑎1 )(𝑥𝑛−𝑘+2 ∨ 𝑎2 ) ⋅ ⋅ ⋅ (𝑥𝑛 ∨
𝑎𝑘 )𝑁 ≰ 𝜙(𝑃).
Since 𝑝 is 𝜙-absorbing primary, 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−𝑘 (𝑥𝑛−𝑘+1 ∨
𝑎1 ) ⋅ ⋅ ⋅ (𝑥𝑛 ∨ 𝑎𝑘 ) ≤ (𝑃 : 1𝑀) or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅
𝑀
𝑥𝑛−𝑘 (𝑥𝑛−𝑘+1 ∨ 𝑎1 )(𝑥𝑛−𝑘+2 ∨ 𝑎2 ) ⋅ ⋅ ⋅ (𝑥𝑛 ∨ 𝑎𝑘 )𝑁 ≤ √
𝑃
for some 𝑖 ∈ {1, 2, . . . , 𝑛 − 1} and so 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1
𝑀
⋅ ⋅ ⋅ 𝑥𝑛−𝑘 𝑥𝑛−𝑘+1 𝑥𝑛−𝑘+2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ √
𝑃 for some 𝑖 ∈ {1, 2, . . . , 𝑛 −
1} or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤ (𝑃 : 1𝑀).
⊆
Similarly, we assume that, for {𝑖1 , 𝑖2 , . . . , 𝑖𝑛−𝑘 }
{1, 2, . . . , 𝑛}, 1 ≤ 𝑘 ≤ 𝑛, 𝑥𝑖1 𝑥𝑖2 ⋅ ⋅ ⋅ 𝑥𝑖𝑛−𝑘 (𝑃 : 1𝑀)𝑘 𝑁 ≤ 𝜙(𝑃).
Also, we assume that 𝑥𝑖1 𝑥𝑖2 ⋅ ⋅ ⋅ 𝑥𝑖𝑛−𝑘 (𝑃 : 1𝑀)𝑘 𝑃 ≤ 𝜙(𝑃),
1 ≤ 𝑘 ≤ 𝑛, because if 𝑥𝑖1 𝑥𝑖2 ⋅ ⋅ ⋅ 𝑥𝑖𝑛−𝑘 (𝑃 : 1𝑀)𝑘 𝑃 ≰ 𝜙(𝑃),
≰
𝜙(𝑃), where
then 𝑥𝑖1 𝑥𝑖2 ⋅ ⋅ ⋅ 𝑥𝑖𝑛−𝑘 𝑎1 𝑎2 ⋅ ⋅ ⋅ 𝑎𝑘 𝑃
𝑎1 𝑎2 𝑎3 ⋅ ⋅ ⋅ 𝑎𝑘 ≤ (𝑃 : 1𝑀) and so 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−𝑘 (𝑥𝑛−𝑘+1 ∨
𝑀
𝑎1 )(𝑥𝑛−𝑘+2 ∨ 𝑎2 ) ⋅ ⋅ ⋅ (𝑥𝑛 ∨ 𝑎𝑘 )(𝑁 ∨ 𝑃) ≤ √
𝑃 and
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−𝑘 (𝑥𝑛−𝑘+1 ∨ 𝑎1 )(𝑥𝑛−𝑘+2 ∨ 𝑎2 ) ⋅ ⋅ ⋅ (𝑥𝑛 ∨ 𝑎𝑘 )(𝑁 ∨ 𝑃) ≰
𝜙(𝑃).
Now, since 𝑃 is 𝜙-absorbing primary, 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤
𝑀
(𝑃 : 1𝑀) or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ √
𝑃, for some 𝑖 ∈
{1, 2, . . . , 𝑛 − 1}.
Also, (𝑃 : 1𝑀)𝑛 𝑃 ≰ 𝜙(𝑃) implies that there exist
𝑎1 , 𝑎2 , . . . , 𝑎𝑛 ≤ (𝑃 : 1𝑀) with 𝑎1 𝑎2 ⋅ ⋅ ⋅ 𝑎𝑛 𝑃 ≰ 𝜙(𝑃). We have
(𝑥1 ∨ 𝑎1 )(𝑥2 ∨ 𝑎2 ) ⋅ ⋅ ⋅ (𝑥𝑛 ∨ 𝑎𝑛 )(𝑁 ∨ 𝑃) ≤ 𝑃 and (𝑥1 ∨ 𝑎1 )(𝑥2 ∨
𝑎2 ) ⋅ ⋅ ⋅ (𝑥𝑛 ∨𝑎𝑛 )(𝑁∨𝑃) ≰ 𝜙(𝑃). Now, 𝑃 is 𝜙-absorbing primary;
therefore, 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛−𝑘 𝑥𝑛−𝑘+1 𝑥𝑛−𝑘+2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃,
for some 𝑖 ∈ {1, 2, . . . , 𝑛 − 1} or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤ (𝑃 : 1𝑀) and
consequently 𝑃 is 𝜙-absorbing primary.
From the above theorem, it follows that if 𝑃 is a 𝜙-primary
element of 𝑀 that is not 𝑛-absorbing primary, then (𝑃 :
1𝑀)𝑛 𝑃 ≤ 𝜙(𝑃).
Corollary 7. Let 𝑀 be a lattice module over a 𝐶-lattice 𝐿. If
𝑃 ∈ 𝑀 is weakly 𝑛-absorbing primary that is not 𝑛-absorbing
primary, then (𝑃 : 1𝑀)𝑛 𝑃 = 0.
Theorem 8. Let 𝑀 be a multiplication lattice module over 𝐶lattice 𝐿 and let 𝑃 ∈ 𝑀. Then the following holds.
(1) Let 𝜓1 , 𝜓2 : 𝑀 → 𝑀 be two functions with 𝜓1 ≤ 𝜓2 ,
that is, 𝜓1 (𝑁) ≤ 𝜓2 (𝑁) for each 𝑁 ∈ 𝑀. Then 𝑃 is
𝜓2 -absorbing primary if it is 𝜓1 -absorbing primary.
(2) Consider the following statements.
(a) 𝑃 is 𝑛-absorbing primary.
(b) 𝑃 is weakly 𝑛-absorbing primary.
(c) 𝑃 is 𝜔-absorbing primary.
(d) 𝑃 is 𝑚-almost 𝑛-absorbing primary.
(e) 𝑃 is almost 𝑛-absorbing primary.
Then (a) ⇒ (b) ⇒ (c) ⇒ (d) ⇒ (e).
(3) 𝑃 is 𝜔-absorbing primary if and only if it is 𝑚-almost
𝑛-absorbing primary for 𝑚 ≥ 1.
Proof. (1) Suppose that 𝑃 ∈ 𝑀 is 𝜓1 -absorbing primary and
also suppose that 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃 and 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰ 𝜓2 (𝑃)
for 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ∈ 𝐿 and 𝑁 ∈ 𝑀. Since 𝜓1 (𝑁) ≤ 𝜓2 (𝑁)
for each 𝑁 ∈ 𝑀, we have 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰ 𝜓1 (𝑃). It follows
from the fact 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃, 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰ 𝜓1 (𝑃), and
𝑃 is 𝜓1 -absorbing primary that 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤ (𝑃 : 1𝑀) or
𝑀
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ √
𝑃 for some 𝑖 ∈ {1, 2, . . . , 𝑛} and
so, 𝑃 is 𝜓2 -absorbing primary.
(2) By definition, every 𝑛-absorbing primary element is
weakly 𝑛-absorbing primary and therefore (a) ⇒ (b) holds.
(b) ⇒ (c) Suppose that 𝑃 is weakly 𝑛-absorbing primary
and also suppose that 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃 and 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰
𝑛
∧∞
𝑛=1 (𝑃 : 1𝑀 ) 𝑃, for 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ∈ 𝐿 and 𝑁 ∈ 𝑀. Then
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≠ 0𝑀. By the assumption, 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤ (𝑃 : 1𝑀)
𝑀
or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ √
𝑃 for some 𝑖 ∈ {1, 2, . . . , 𝑛}
and so, 𝑃 is 𝜔-absorbing primary.
(c) ⇒ (d) Suppose that 𝑃 is 𝜔-absorbing primary and
also suppose that 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃 and 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰ (𝑃 :
1𝑀)𝑚 𝑃 for 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ∈ 𝐿, 𝑁 ∈ 𝑀, and 𝑚 ≥ 1. Then
𝑚
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃 and 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰ ∧∞
𝑚=1 (𝑃 : 1𝑀 ) 𝑃.
Since 𝑃 is 𝜔-absorbing primary, it follows that 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤
𝑀
(𝑃 : 1𝑀) or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ √
𝑃 for some 𝑖 ∈
{1, 2, . . . , 𝑛} and so, 𝑃 is 𝑚-almost 𝑛-absorbing primary.
The statement (e) is a particular case of the statement (d)
for 𝑚 = 1 and therefore (d) ⇒ (e) holds.
(3) Suppose that 𝑃 ∈ 𝑀 is 𝑚-almost 𝑛-absorbing primary
for 𝑚 ≥ 1 and also suppose that 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃 and
𝑚
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰ ∧∞
𝑚=1 (𝑃 : 1𝑀 ) 𝑃 for 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ∈ 𝐿
and 𝑁 ∈ 𝑀. Then 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑚 𝑁 ≤ 𝑃 and 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰
(𝑃 : 1𝑀)𝑚 𝑃 for some 𝑚 ≥ 1. Since 𝑃 is 𝑚-almost 𝑛absorbing primary, we have 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤ (𝑃 : 1𝑀) or
𝑀
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ √
𝑃 for some 𝑖 ∈ {1, 2, . . . , 𝑛}.
Consequently, 𝑃 is 𝜔-absorbing primary.
The converse follows from (c) ⇒ (d).
Corollary 9. Let 𝑀 be a lattice module over a 𝐶-lattice 𝐿. If
𝑃 ∈ 𝑀 is 𝜙-absorbing primary, where 𝜙 ≤ 𝜙𝑛+2 , then 𝑃 is 𝜔absorbing primary.
Proof. Suppose that 𝑃 ∈ 𝑀 is 𝜙-absorbing primary. If 𝑃 is
𝑛-absorbing primary, then, by Theorem 8, it is 𝜔-absorbing
primary. Now, if 𝑃 is not 𝑛-absorbing primary, then, by
Theorem 6, (𝑃 : 1𝑀)𝑛 𝑃 ≤ 𝜙(𝑃) ≤ 𝜙𝑛+2 (𝑃) = (𝑃 : 1𝑀)𝑛+1 𝑃.
Consequently, 𝜙(𝑃) = (𝑃 : 1𝑀)𝑘 𝑃, for each 𝑘 ≥ 𝑛. And, by
Theorem 8(3), 𝑃 is 𝜔-absorbing primary.
Lemma 10. Let 𝐿 be a multiplicative lattice. If 𝑝 ∈ 𝐿 is 𝜙absorbing primary with 𝜙(√𝑝) = √𝜙(𝑝), then √𝑝 is also 𝜙absorbing.
≤
Proof. (I) Suppose that 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛+1
√𝑝 and
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛+1 ≰ 𝜙(√𝑝) = √𝜙(𝑝), for 𝑥1 , 𝑥2 , . . . , 𝑥𝑛+1 ∈ 𝐿.
If 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛+1 ≰ √𝑝 for 𝑖 ∈ {1, 2, . . . , 𝑛}, then we
have (𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛+1 )𝑠 ≰ 𝑝 for any positive integer
𝑠 and for 𝑖 ∈ {1, 2, . . . , 𝑛}.
(II) Now, since 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛+1 ≤ √𝑝, there exists a positive
𝑘
integer 𝑘 such that (𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛+1 )𝑘 = 𝑥1𝑘 𝑥2𝑘 ⋅ ⋅ ⋅ 𝑥𝑛+1
≤ 𝑝.
4
Algebra
𝑡
(III) As 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛+1 ≰ √𝜙(𝑝), we have 𝑥1𝑡 𝑥2𝑡 ⋅ ⋅ ⋅ 𝑥𝑛+1
≰
𝜙(𝑝) for a positive integer 𝑡.
From the assumption and (I), (II), and (III), it follows
that (𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 )𝑘 = 𝑥1𝑘 𝑥2𝑘 ⋅ ⋅ ⋅ 𝑥𝑛𝑘 ≤ √𝑝. Consequently,
𝑀2
1
⇒ 𝑆1 ≤ 𝑀√𝑃
1 and 𝑆2 ≤ √𝑃2 ;
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤ √√𝑝 and therefore √𝑝 is 𝜙-absorbing primary.
𝑥 ≤√(𝑃1 : 1𝑀1 ) and 𝑦 ≤√(𝑃2 : 1𝑀2 );
It is easy to observe that if (𝐿 1 , ∧1 , ∨1 , ∘1 ) and (𝐿 2 , ∧2 ,
∨2 , ∘2 ) are multiplicative lattices then 𝐿 1 × 𝐿 2 is also a
multiplicative lattice with componentwise meet, join, and
multiplication.
Also, if 𝑀1 and 𝑀2 are lattice modules over multiplicative
lattices 𝐿 1 and 𝐿 2 , respectively, then 𝑀1 × 𝑀2 is a lattice
module over 𝐿 1 × 𝐿 2 with componentwise meet, join, and
multiplication given by (𝑎, 𝑏)(𝑁1 , 𝑁2 ) = (𝑎𝑁1 , 𝑏𝑁2 ), where
(𝑎, 𝑏) ∈ 𝐿 1 × 𝐿 2 and (𝑁1 , 𝑁2 ) ∈ 𝑀1 × 𝑀2 .
Lemma 11. Let 𝑀 = 𝑀1 × 𝑀2 and 𝐿 = 𝐿 1 × 𝐿 2 , where 𝑀𝑖
is a lattice module over 𝐶-lattice 𝐿 𝑖 , for 𝑖 = 1, 2. Then ((𝑃1 :
1𝑀1 ), (𝑃2 : 1𝑀2 )) = ((𝑃1 , 𝑃2 ) : (1𝑀1 , 1𝑀2 )), for 𝑃1 ∈ 𝑀1 and
𝑃2 ∈ 𝑀2 .
Proof. Let (𝑥, 𝑦) ∈ 𝐿 ∗ . Now, (𝑥, 𝑦) ≤ ((𝑃1 : 1𝑀1 ), (𝑃2 : 1𝑀2 )):
⇔ 𝑥 ≤ (𝑃1 : 1𝑀1 ) and 𝑦 ≤ (𝑃2 : 1𝑀2 ),
⇒ 𝑆1 ≤ 𝑥1𝑀1 and 𝑆2 ≤ 𝑦1𝑀2 for some 𝑥, 𝑦 ∈ 𝐿 with
⇒ 𝑥𝑡 ≤ (𝑃1 : 1𝑀1 ) and 𝑦𝑟 ≤ (𝑃2 : 1𝑀2 ) for some
positive integers 𝑡 and 𝑟.
Choose 𝑘 = 𝑡 + 𝑟. Then 𝑥𝑘 ≤ (𝑃1 : 1𝑀1 ) and 𝑦𝑘 ≤ (𝑃2 : 1𝑀2 ):
⇒ (𝑥, 𝑦)𝑘 = (𝑥𝑘 , 𝑦𝑘 ) ≤ ((𝑃1 : 1𝑀1 ), (𝑃2 : 1𝑀2 ));
⇒ (𝑥, 𝑦) ≤√((𝑃1 : 1𝑀1 ), (𝑃2 : 1𝑀2 ));
⇒ (𝑥, 𝑦)(1𝑀1 , 1𝑀2 ) ≤ √((𝑃1 : 1𝑀1 ), (𝑃2 : 1𝑀2 )) (1𝑀1 ,
1𝑀2 );
⇒
(𝑆1 , 𝑆2 )
≤
(𝑥, 𝑦)(1𝑀1 , 1𝑀2 )
≤
√((𝑃1 : 1𝑀1 ), (𝑃2 : 1𝑀2 ))(1𝑀1 , 1𝑀2 ) = √(𝑃1 , 𝑃2 ).
𝑀
Consequently,
𝑀
1
2
𝑃1 , 𝑀√
𝑃2 ) ≤ √
(𝑃1 , 𝑃2 ).
(𝑀√
(II)
⇔ 𝑥1𝑀1 ≤ 𝑃1 and 𝑦1𝑀2 ≤ 𝑃2 ,
From (I) and (II), result follows.
⇔ (𝑥, 𝑦)(1𝑀1 , 1𝑀2 ) = (𝑥1𝑀1 , 𝑦1𝑀2 ) ≤ (𝑃1 , 𝑃2 ),
Theorem 13. Let 𝑀 = 𝑀1 × 𝑀2 and 𝐿 = 𝐿 1 × 𝐿 2 , where 𝑀𝑖 is
a lattice module over 𝐶-lattice 𝐿 𝑖 , for 𝑖 = 1, 2, and let 𝜙 : 𝑀 →
𝑀 be a function. If 𝑃1 ∈ 𝑀1 is a weakly 𝑛-absorbing primary
such that (0, 1𝑀2 ) ≤ 𝜙(𝑃1 , 1𝑀2 ), then (𝑃1 , 1𝑀2 ) ∈ 𝑀1 × 𝑀2 is
𝜙-absorbing primary.
⇔ (𝑥, 𝑦) ≤ (𝑃1 , 𝑃2 ) : (1𝑀1 , 1𝑀2 ).
Lemma 12. Let 𝑀 = 𝑀1 × 𝑀2 and 𝐿 = 𝐿 1 × 𝐿 2 , where 𝑀𝑖 is a
𝑀
lattice module over 𝐶-lattice 𝐿 𝑖 , for 𝑖 = 1, 2. Then √(𝑃
1 , 𝑃2 ) =
𝑀1
𝑀2
( √𝑃1 , √𝑃2 ), for 𝑃1 ∈ 𝑀1 and 𝑃2 ∈ 𝑀2 .
𝑀
Proof. Let (𝑁1 , 𝑁2 ) ∈ 𝑀∗ with (𝑁1 , 𝑁2 ) ≤ √(𝑃
1 , 𝑃2 ) =
≤
√(𝑃1 , 𝑃2 ) : (1𝑀1 , 1𝑀2 )(1𝑀1 , 1𝑀2 ). Then (𝑁1 , 𝑁2 )
(𝑎, 𝑏)(1𝑀1 , 1𝑀2 ), where (𝑎, 𝑏) ≤√(𝑃1 , 𝑃2 ) : (1𝑀1 , 1𝑀2 ).
Now, (𝑎, 𝑏) ≤√(𝑃1 , 𝑃2 ) : (1𝑀1 , 1𝑀2 ) ⇒ (𝑎, 𝑏)𝑛 ≤ (𝑃1 , 𝑃2 ) :
(1𝑀1 , 1𝑀2 ) for a positive integer 𝑛:
⇒ 𝑎𝑛 ≤ (𝑃1 : 1𝑀1 ) and 𝑏𝑛 ≤ (𝑃2 : 1𝑀2 ) (by Lemma 11),
⇒ (𝑎, 𝑏) ≤ (√(𝑃1 : 1𝑀1 ),√(𝑃2 : 1𝑀2 )),
⇔ (𝑎, 𝑏)(1𝑀1 , 1𝑀2 ) ≤ (√(𝑃1 : 1𝑀1 ),√(𝑃2 : 1𝑀2 )) (1𝑀1 ,
1𝑀2 ) = (√(𝑃1 : 1𝑀1 )1𝑀1 ,√(𝑃2 : 1𝑀2 )1𝑀2 ),
𝑀2
1
⇔ (𝑎, 𝑏)(1𝑀1 , 1𝑀2 ) ≤ (𝑀√𝑃
1 , √𝑃2 ).
Consequently,
𝑀
1
2
√
(𝑃1 , 𝑃2 ) ≤ (𝑀√
𝑃1 , 𝑀√
𝑃2 ) .
(I)
1
Next, let (𝑆1 , 𝑆2 ) ∈ 𝑀∗ be such that (𝑆1 , 𝑆2 ) ≤ (𝑀√𝑃
1,
√𝑃2 ):
𝑀2
⇒ 𝑆1 ≤√(𝑃1 : 1𝑀1 )1𝑀1 and 𝑆2 ≤√(𝑃2 : 1𝑀2 )1𝑀2 ;
Proof. Let (𝑎1 , 𝑏1 ), (𝑎2 , 𝑏2 ), . . . , (𝑎𝑛 , 𝑏𝑛 ) ∈ 𝐿 and (𝑁1 , 𝑁2 ) ∈ 𝑀
=
be such that (𝑎1 , 𝑏1 )(𝑎2 , 𝑏2 ) ⋅ ⋅ ⋅ (𝑎𝑛 , 𝑏𝑛 )(𝑁1 , 𝑁2 )
(𝑎1 𝑎2 ⋅ ⋅ ⋅ 𝑎2 𝑁1 , 𝑏1 𝑏2 ⋅ ⋅ ⋅ 𝑏2 𝑁2 ) ≤ (𝑃1 , 1𝑀2 ) and (𝑎1 , 𝑏1 )(𝑎2 , 𝑏2 )
⋅ ⋅ ⋅ (𝑎𝑛 , 𝑏𝑛 )(𝑁1 , 𝑁2 ) = (𝑎1 𝑎2 ⋅ ⋅ ⋅ 𝑎𝑛 𝑁1 , 𝑏1 𝑏2 ⋅ ⋅ ⋅ 𝑏𝑛 𝑁2 ) ≰ 𝜙(𝑃1 ,
1𝑀2 ).
Since (0, 1𝑀2 ) ≤ 𝜙(𝑃1 , 1𝑀2 ), we have (𝑎1 , 𝑏1 )(𝑎2 , 𝑏2 ) ⋅ ⋅ ⋅
(𝑎𝑛 , 𝑏𝑛 )(𝑁1 , 𝑁2 ) = (𝑎1 𝑎2 ⋅ ⋅ ⋅ 𝑎𝑛 𝑁1 , 𝑏1 𝑏2 ⋅ ⋅ ⋅ 𝑏𝑛 𝑁2 ) ≰ (0, 1𝑀2 )
and so 0 ≠ 𝑎1 𝑎2 ⋅ ⋅ ⋅ 𝑎𝑛 𝑁1 ≤ 𝑃1 . Since 𝑃1 ∈ 𝑀1
is weakly 𝑛-absorbing primary, we have 𝑎1 𝑎2 ⋅ ⋅ ⋅ 𝑎𝑛 ≤
1
(𝑃1 : 1𝑀1 ) or 𝑎1 𝑎2 ⋅ ⋅ ⋅ 𝑎𝑖−1 𝑎𝑖+1 ⋅ ⋅ ⋅ 𝑎𝑛 𝑁1 ≤ 𝑀√𝑃
1 , for
some 𝑖 ∈ {1, 2, . . . , 𝑛}. This implies that (𝑎1 , 𝑏1 )(𝑎2 ,
𝑏2 ) ⋅ ⋅ ⋅ (𝑎𝑛 , 𝑏𝑛 ) = (𝑎1 𝑎2 ⋅ ⋅ ⋅ 𝑎𝑛 , 𝑏1 𝑏2 ⋅ ⋅ ⋅ 𝑏𝑛 ) ≤ ((𝑃1 : 1𝑀1 ),
(1𝑀2 : 1𝑀2 )) = ((𝑃1 , 1𝑀2 ) : (1𝑀1 , 1𝑀2 )) or (𝑎1 , 𝑏1 )(𝑎2 , 𝑏2 )
1
⋅ ⋅ ⋅ (𝑎𝑖−1 , 𝑏𝑖−1 )(𝑎𝑖+1 , 𝑏𝑖+1 ) ⋅ ⋅ ⋅ (𝑎𝑛 , 𝑏𝑛 )(𝑁1 , 𝑁2 ) ≤ (𝑀√𝑃
1 , 1𝑀2 ) =
𝑀
√(𝑃
1 , 1𝑀2 ) for some 𝑖 ∈ {1, 2, . . . , 𝑛}, by Lemmas 11 and
12. Therefore, (𝑃1 , 1𝑀2 ) is 𝜙-absorbing primary element of
𝑀1 × 𝑀2 .
Theorem 14. Let 𝑀 be a lattice module over a 𝐶-lattice 𝐿
and 𝑃 ∈ 𝑀. Then 𝑃 is 𝜙-absorbing primary if and only
if (𝑃 : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) = (𝑃 : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 1𝑀) or
(𝑃 : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) = (𝜙(𝑃) : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) or (𝑃 :
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) = (√𝑃 : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) for some
𝑖 ∈ {1, 2, . . . , 𝑛 − 1}, for 𝑥1 , 𝑥2 , . . . , 𝑥𝑛−1 ∈ 𝐿 and 𝑁 ∈ 𝑀 with
𝑀
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁 ≰ √
𝑃.
Algebra
Proof. Suppose that 𝑃 ∈ 𝑀 is 𝜙-absorbing primary and
𝑀
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁 ≰ √
𝑃, for 𝑥1 , 𝑥2 , . . . , 𝑥𝑛−1 ∈ 𝐿 and 𝑁 ∈ 𝑀. Let
𝑟 ∈ 𝐿 ∗ be such that 𝑟 ≤ (𝑃 : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) which essentially
implies that 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑟𝑁 ≤ 𝑃.
We have the following two cases.
Case 1. If 𝑟𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁 ≰ 𝜙(𝑃), then as 𝑃 is 𝜙absorbing primary, we have 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑟 ≤ (𝑃 : 1𝑀) or
𝑀
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑟𝑁 ≤ √
𝑃 for some 𝑖 ∈ {1, 2, . . . , 𝑛 −
𝑀
1}. Therefore, 𝑟 ≤ (𝑃 : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 1𝑀) or 𝑟 ≤ (√
𝑃 :
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) for some 𝑖 ∈ {1, 2, . . . , 𝑛 − 1}.
Case 2. If 𝑟𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁 ≤ 𝜙(𝑃), then 𝑟 ≤ (𝜙(𝑃) :
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁). So (𝑃 : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) ≤ (𝜙(𝑃) :
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁).
Now, from Cases 1 and 2, it follows that (𝑃 :
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) = (𝑃 : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 1𝑀) or (𝑃 : 𝑥1 𝑥2 ⋅ ⋅ ⋅
𝑥𝑛−1 𝑁) = (𝜙(𝑃) : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) or (𝑃 : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) =
𝑀
(√
𝑃 : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) for some 𝑖 ∈ {1, 2, . . . , 𝑛−1}.
Conversely, suppose that 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃 and
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰ 𝜙(𝑃), for 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ∈ 𝐿 and 𝑁 ∈ 𝑀. If
𝑀
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁 ≤ √
𝑃, then the result is obvious. So, suppose
𝑀
that 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁 ≰ √
𝑃.
Now, (𝑃 : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) = (𝑃 : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 1𝑀) or
(𝑃 : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) = (𝜙(𝑃) : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) or (𝑃 :
𝑀
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) = (√
𝑃 : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) for some
𝑖 ∈ {1, 2, . . . , 𝑛 − 1}. Since 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑃, we have 𝑥𝑛 ≤
(𝑝 : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁). But 𝑥𝑛 ≰ (𝜙(𝑃) : 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑁) and so
𝑀
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛−1 𝑥𝑛 ≤ (𝑃 : 1𝑀) or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ √
𝑃
for some 𝑖 ∈ {1, 2, . . . , 𝑛}. Consequently, 𝑃 is 𝜙-absorbing
primary.
Theorem 15. Let 𝑀 = 𝑀1 × 𝑀2 × ⋅ ⋅ ⋅ × 𝑀𝑛 and 𝐿 = 𝐿 1 × 𝐿 2 ×
⋅ ⋅ ⋅×𝐿 𝑛 , where each 𝑀𝑖 is a compactly generated lattice module
over a 𝐶-lattice 𝐿 𝑖 , for 𝑖 ∈ {1, 2, . . . , 𝑛}. Let 𝜓 : 𝑀 → 𝑀
such that 𝜓(𝑃) = (𝜓1 (𝑃1 ), 𝜓2 (𝑃2 ), . . . , 𝜓𝑛 (𝑃𝑛 )), where 𝑃𝑖 ∈ 𝑀𝑖 ,
𝜓𝑖 : 𝑀𝑖 → 𝑀𝑖 , 𝑖 ∈ {1, 2, . . . , 𝑛}, and 𝑃 = (𝑃1 , 𝑃2 , . . . , 𝑃𝑛 ) is 𝜓absorbing primary. Then 𝑃𝑖 is a 𝜓𝑖 -absorbing primary element
of 𝑀𝑖 , for each 𝑖 with 𝑃𝑖 ≠ 1𝑀𝑖 .
Proof. Let 𝑃𝑖 ≠ 1𝑀𝑖 , 𝑁𝑖 ∈ 𝑀𝑖 , and 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ∈ 𝐿 𝑖 be
such that 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁𝑖 ≤ 𝑃𝑖 and 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁𝑖 ≰ 𝜓𝑖 (𝑃𝑖 ). Thus
(1, . . . , 1, 𝑥1 , 1, . . . , 1)(1, . . . , 1, 𝑥2 , 1, . . . , 1) ⋅ ⋅ ⋅ (1, . . . , 1, 𝑥𝑛 , 1,
=
(0, . . . , 0, 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁𝑖 ,
. . . , 1)(0, . . . , 0, 𝑁𝑖 , 0, . . . , 0)
0, . . . , 0) ≤ 𝑃, and (0, . . . , 0, 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁𝑖 , 0, . . . , 0) ≰ 𝜓(𝑃).
As 𝑃 is 𝜓-absorbing primary, (1, 1, . . . , 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 , 0, . . . , 0) ≤
(𝑃 : 1𝑀) or (1, 1, . . . , 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁𝑖 , 0, . . . , 0) ≤
𝑀
√
𝑃. Now, by Lemmas 11 and 12, we have 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤ (𝑃𝑖 :
𝑀
1𝑀𝑖 ) or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁𝑖 ≤ √𝑃
𝑖 for 𝑖 ∈ {1, 2, . . . , 𝑛}
and consequently, 𝑃𝑖 is 𝜓𝑖 -absorbing primary element of 𝑀𝑖 ,
for each 𝑖.
Corollary 16. Let 𝑀 = 𝑀1 × 𝑀2 × ⋅ ⋅ ⋅ × 𝑀𝑛 and 𝐿 = 𝐿 1 × 𝐿 2 ×
⋅ ⋅ ⋅×𝐿 𝑛 , where each 𝑀𝑖 is a compactly generated lattice module
over a 𝐶-lattice 𝐿 𝑖 , for 𝑖 ∈ {1, 2, . . . , 𝑛}. If 𝑃 = (𝑃1 , 𝑃2 , . . . , 𝑃𝑛 ) ∈
𝑀 is 𝜙𝑚 -absorbing primary, where 𝑃𝑖 ∈ 𝑀𝑖 , then 𝑃𝑖 ∈ 𝑀𝑖 is
𝜙𝑚 -absorbing primary with 𝑃𝑖 ≠ 1𝑀𝑖 (𝑛, 𝑚 ≥ 2).
5
Proof. We have 𝜙𝑚 (𝑃) = (𝑃 : 1𝑀)𝑚−1 𝑃 = ((𝑃1 :
1𝑀1 )𝑚−1 𝑃1 , (𝑃2 : 1𝑀2 )𝑚−1 𝑃2 , . . . , (𝑃𝑛 : 1𝑀𝑛 )𝑚−1 𝑃𝑛 ) =
(𝜙𝑚 (𝑃1 ), 𝜙𝑚 (𝑃2 ), . . . , 𝜙𝑚 (𝑃𝑛 )) and the result follows from
Theorem 15.
Theorem 17. Let 𝑀 be a lattice module over a 𝐶-lattice 𝐿.
Suppose that 𝑎1𝑀 ∈ 𝑀 is a weak join principal element with
𝑎1𝑀 ≠ 1𝑀 and (0𝑀 : 𝑎1𝑀) ≤ √𝑎. Then 𝑎1𝑀 is 𝑚-almost
𝑛-absorbing primary, 𝑚 ≥ 1, if and only if it is 𝑛-absorbing
primary.
Proof. Suppose that 𝑎1𝑀 ∈ 𝑀 is 𝑚-almost 𝑛-absorbing
primary and 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑎1𝑀, for 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ∈ 𝐿
and 𝑁 ∈ 𝑀. If 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰ (𝑎1𝑀 : 1𝑀)𝑚 𝑎1𝑀, then
𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤ (𝑎1𝑀 : 1𝑀) or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤
𝑀
√𝑎1
𝑀 for some 𝑖 ∈ {1, 2, . . . , 𝑛} and therefore 𝑎1𝑀 is 𝑛absorbing primary.
Now, suppose that 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ (𝑎1𝑀 : 1𝑀)𝑚 𝑎1𝑀.
Since 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑎1𝑀, we have (𝑥1 ∨ 𝑎)𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑎1𝑀.
If (𝑥1 ∨ 𝑎)𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰ (𝑎1𝑀 : 1𝑀)𝑚 𝑎1𝑀, then it follows
from the fact (𝑥1 ∨ 𝑎)𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ 𝑎1𝑀, (𝑥1 ∨ 𝑎)𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≰
(𝑎1𝑀 : 1𝑀)𝑚 𝑎1𝑀, and 𝑎1𝑀 is 𝑚-almost 𝑛-absorbing primary
that 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤ (𝑎1𝑀 : 1𝑀) or 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑖−1 𝑥𝑖+1 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤
𝑀
√𝑎1
𝑀 for some 𝑖 ∈ {1, 2, . . . , 𝑛} and we are done.
So assume that (𝑥1 ∨ 𝑎)𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ (𝑎1𝑀 : 1𝑀)𝑚 𝑎1𝑀.
Then as 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ (𝑎1𝑀 : 1𝑀)𝑛 𝑎1𝑀, we have
𝑎𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ (𝑎1𝑀 : 1𝑀)𝑛 𝑎1𝑀. Next, 𝑎𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤ (𝑎1𝑀 :
1𝑀)𝑛 𝑎1𝑀 and 𝑎1𝑀 is weak join principal; together they imply
that 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 ≤ ((𝑎1𝑀 : 1𝑀)𝑛 𝑎1𝑀 : 𝑎1𝑀) = (𝑎1𝑀 :
1𝑀)𝑛 ∨ (0𝑀 : 𝑎1𝑀) ≤ 𝑎 ∨ (0𝑀 : 𝑎1𝑀) ≤ √𝑎. Consequently,
𝑀
𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑁 ≤√𝑎𝑁 ≤√𝑎1𝑀 = √𝑎1
𝑀 which implies that 𝑎1𝑀
is 𝑛-absorbing primary.
The converse follows from Theorem 8(2).
Note. The results pertaining to 𝜙-absorbing elements are
essentially the corollaries to the respective results of 𝜙absorbing primary elements, as such the results of 𝜙absorbing elements are the immediate consequences of
results of 𝜙-absorbing primary elements in this paper.
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.
Acknowledgment
This research work is an outcome of the project supported
by Board of College and University Development, Savitribai
Phule Pune University, Pune.
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