Research Article
On -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; ballalshyam@gmail.com
Received 18 December 2014; Accepted 31 March 2015
Academic Editor: Andrei V. Kelarev
Copyright © 2015 S. Ballal and V. Kharat. Tis 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 be a -lattice and let be a lattice module over . Let :→ be a function. A proper element ∈ is said
to be -absorbing primary if, for
1
,
2
,...,
∈ and ∈,
1
2
⋅⋅⋅
≤ and
1
2
⋅⋅⋅
≰ () together imply
1
2
⋅⋅⋅
≤(:1
) or
1
2
⋅⋅⋅
−1
+1
⋅⋅⋅
≤
√
, for some ∈{1,2,...,}. We study some basic properties of -absorbing
primary elements. Also, various generalizations of prime and primary elements in multiplicative lattices and lattice modules as
-absorbing elements and -absorbing primary elements are unifed.
1. Introduction
A lattice is said to be complete, if, for any subset of ,
we have ∨,∧∈. Since every complete lattice is bounded,
1
(or 1) denotes the greatest element and 0
(or 0) denotes
the smallest element of . A complete lattice is said to be a
multiplicative lattice, if there is a defned binary operation “⋅”
called multiplication on satisfying the following conditions:
(1) ⋅=⋅, for all ,∈,
(2) ⋅(⋅)=(⋅)⋅, for all ,,∈,
(3) ⋅(∨
)=∨
(⋅
), for all ,
∈,
(4) ⋅1=, for all ∈.
Henceforth, ⋅ will be simply denoted by .
An element ̸ =1 of a multiplicative lattice is said to be
prime if ≤ implies either ≤ or ≤, for ,∈.
Radical of an element ∈ is denoted by√ and is defned
as √ =∨{∈|
≤, for some ∈ Z
+
}.
An element of a complete lattice is said to be compact
if ≤∨
implies ≤∨
=1
, where ∈ Z
+
. Te set of all
compact elements of a lattice is denoted by
∗
. By a -lattice
we mean a multiplicative lattice with a multiplicatively
closed set of compact elements which generates under
join.
A complete lattice is said to be a lattice module over
a multiplicative lattice , if there is a multiplication between
elements of and , denoted by for ∈ and ∈,
which satisfes the following properties:
(1) ()=(),
(2) (∨
)(∨
)=∨
,
(
),
(3) 1
=,
(4) 0
=0
, for all ,,
∈ and for all ,
∈,
where 1
denotes the greatest element of and 0
denotes
the smallest element of .
For ∈ and ∈, denote ( : ) = ∨{ ∈ :
≤ }. For ,∈, ( : ) = ∨{ ∈ : ≤ } and
for ,∈, (:)=∨{∈:≤}. For ∈,
√
=∨{∈:
1
≤ } for some positive integer
and it is also denoted by√ (:1
). For ∈, we defne
√ =√ (:1
)1
. An element ∈ is said to be weak
join principal if it satisfes the following identity ∨(0
:)=
(:) for all ∈.
A lattice module over a multiplicative lattice is called
a multiplication lattice module if for ∈ there exists an
element ∈ such that =1
.
An element ̸ =1
in is said to be prime if ≤
implies ≤ or 1
≤, that is, ≤(:1
) for every
∈ and ∈.
An element ∈ is called compact if ≤∨
implies
≤
1
∨
2
∨⋅⋅⋅∨
for some
1
,
2
,...,
. If each
element of is a join of principal (compact) elements of
Hindawi Publishing Corporation
Algebra
Volume 2015, Article ID 183930, 6 pages
http://dx.doi.org/10.1155/2015/183930