# Onϕ-Absorbing Primary Elements in Lattice Modules

Authors: Sachin Ballal; Vilas Kharat
Journal: Algebra
 (2015)

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## Abstract

LetL 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, x1x2⋯xnN≤P and x1x2⋯xnN≰ϕ(P) together imply x1x2⋯xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤PM, for some i∈{1,2,…,n}. We study some basic properties of ϕ-absorbing 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.


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## Body

## 1. Introduction

A latticeL is said to becomplete, 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 amultiplicative 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·(∨αbα)=∨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 elementp≠1 of a multiplicative lattice L is said to beprime 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 elementc of a complete lattice L is said to becompact if c≤∨αaα implies c≤∨i=1naαi, 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 latticeM is said to be alattice 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)
∨αaα∨βNβ=∨α,β(aαNβ),(3)
1LN=N,(4)
0LN=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.ForN∈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:xn1M≤N} for some positive integer n and it is also denoted by (N:1M). For N∈M, we define NM=(N:1M)1M. An element N∈M is said to beweak join principal if it satisfies the following identity a∨(0M:N)=(aN:N) for all a∈L.A lattice moduleM over a multiplicative lattice L is called amultiplication lattice module if for N∈M there exists an element a∈L such that N=a1M.An elementN≠1M in M is said to beprime if aX≤N implies X≤N or a1M≤N, that is, a≤(N:1M) for every a∈L and X∈M.An elementN∈M is calledcompact if N≤∨αAα implies N≤Aα1∨Aα2∨⋯∨Aαn for some α1,α2,…,αn. If each element of M is a join of principal (compact) elements of M, then M is calledprincipally 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 beweakly prime if 0≠ab≤p implies either a≤p or b≤p andalmost 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 beweakly primary if 0≠ab≤p implies either a≤p or b≤p andalmost 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 beweakly 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, 2-absorbing, weakly 2-absorbing, and almost prime elements, where n≥2. An element p<1 of a multiplicative lattice L is called n-absorbing if x1x2x3⋯xn+1≤p implies x1x2⋯xi-1xi+1⋯xn+1≤p and calledweaklyn-absorbing if, for x1,x2,x3,…,xn+1∈L, 0≠x1x2x3⋯xn+1≤p implies x1x2⋯xi-1xi+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-almostn-absorbing if x1x2x3⋯xn+1≤p and x1x2x3⋯xn+1≰pn together imply x1x2⋯xi-1xi+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].


## 2.ϕ-Absorbing Primary Elements

We introduce the concepts ofϕ-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, whereM is a lattice module over a multiplicative lattice L and n≥1.Definition 1.
A proper elementP∈M is said to be n-absorbing if x1x2⋯xnN≤P implies x1x2⋯xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤P for i∈{1,2,…,n}, where x1,x2,…,xn∈L and N∈M.Definition 2.
Letϕ:M→M be a function. A proper element P∈M is said to be ϕ-absorbing if x1x2⋯xnN≤P and x1x2⋯xnN≰ϕ(P) together imply x1x2⋯xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤P for i∈{1,2,…,n}, where x1,x2,…,xn∈L and N∈M.Definition 3.
A proper elementP∈M is said to be n-absorbing primary if x1x2⋯xnN≤P implies x1x2⋯xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤PM for i∈{1,2,…,n}, where x1,x2,…,xn∈L and N∈M.Definition 4.
Letϕ:M→M be a function. A proper element P∈M is said to be ϕ-absorbing primary if x1x2⋯xnN≤P and x1x2⋯xnN≰ϕ(P) together imply x1x2⋯xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤PM for i∈{1,2,…,n}, where x1,x2,…,xn∈L and N∈M.LetM 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)mP defines m-almost n-absorbing primary elements of M.(4)
ϕω:ϕP=∧m=1∞(P:1M)mP defines ω-absorbing primary elements of M.Remark 5.
It follows immediately from the definition that anyn-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.
LetM be a lattice module over a C-lattice L. Then every ϕ-absorbing primary element P∈M with (P:1M)nP≰ϕ(P) is n-absorbing primary.Proof.
Suppose that(P:1M)nP≰ϕ(P) and x1x2⋯xnN≤P, for x1,x2,…,xn∈L, N∈M. If x1x2⋯xnN≰ϕ(P), then as P is ϕ-absorbing primary, we have x1x2⋯xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤PM for some i∈{1,2,…,n}, which concludes that P is n-absorbing primary.
Now, suppose thatx1x2⋯xnN≤ϕ(P). We assume that x1x2⋯xn-k(P:1M)kN≤ϕ(P), for all k∈{1,2,…,n-1}. If x1x2⋯xn-k(P:1M)kN≰ϕ(P), then there exists a1,a2,…,ak≤(P:1M) such that x1x2⋯xn-ka1a2⋯akN≰ϕ(P). Hence, x1x2⋯xn-k(xn-k+1∨a1)(xn-k+2∨a2)⋯(xn∨ak)N≤P and x1x2⋯xn-k(xn-k+1∨a1)(xn-k+2∨a2)⋯(xn∨ak)N≰ϕ(P).
Sincep is ϕ-absorbing primary, x1x2⋯xn-k(xn-k+1∨a1)⋯(xn∨ak)≤(P:1M) or x1x2⋯xi-1xi+1⋯xn-k(xn-k+1∨a1)(xn-k+2∨a2)⋯(xn∨ak)N≤PM for some i∈{1,2,…,n-1} and so x1x2⋯xi-1xi+1⋯xn-kxn-k+1xn-k+2⋯xnN≤PM for some i∈{1,2,…,n-1} or x1x2⋯xn≤(P:1M).
Similarly, we assume that, for{i1,i2,…,in-k}⊆{1,2,…,n}, 1≤k≤n, xi1xi2⋯xin-k(P:1M)kN≤ϕ(P). Also, we assume that xi1xi2⋯xin-k(P:1M)kP≤ϕ(P), 1≤k≤n, because if xi1xi2⋯xin-k(P:1M)kP≰ϕ(P), then xi1xi2⋯xin-ka1a2⋯akP≰ϕ(P), where a1a2a3⋯ak≤(P:1M) and so x1x2⋯xn-k(xn-k+1∨a1)(xn-k+2∨a2)⋯(xn∨ak)(N∨P)≤PM and x1x2⋯xn-k(xn-k+1∨a1)(xn-k+2∨a2)⋯(xn∨ak)(N∨P)≰ϕ(P).
Now, sinceP is ϕ-absorbing primary, x1x2⋯xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤PM, for some i∈{1,2,…,n-1}.
Also,(P:1M)nP≰ϕ(P) implies that there exist a1,a2,…,an≤(P:1M) with a1a2⋯anP≰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, x1x2⋯xi-1xi+1⋯xn-kxn-k+1xn-k+2⋯xnN≤P, for some i∈{1,2,…,n-1} or x1x2⋯xn≤(P:1M) and consequently P is ϕ-absorbing primary.From the above theorem, it follows that ifP is a ϕ-primary element of M that is not n-absorbing primary, then (P:1M)nP≤ϕ(P).Corollary 7.
LetM 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)nP=0.Theorem 8.
LetM be a multiplication lattice module over C-lattice L and let P∈M. Then the following holds.(1)
Letψ1,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 thatP∈M is ψ1-absorbing primary and also suppose that x1x2⋯xnN≤P and x1x2⋯xnN≰ψ2(P) for x1,x2,…,xn∈L and N∈M. Since ψ1(N)≤ψ2(N) for each N∈M, we have x1x2⋯xnN≰ψ1(P). It follows from the fact x1x2⋯xnN≤P, x1x2⋯xnN≰ψ1(P), and P is ψ1-absorbing primary that x1x2⋯xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤PM for some i∈{1,2,…,n} and so, P is ψ2-absorbing primary.
(2) By definition, everyn-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 x1x2⋯xnN≤P and x1x2⋯xnN≰∧n=1∞(P:1M)nP, for x1,x2,…,xn∈L and N∈M. Then x1x2⋯xnN≠0M. By the assumption, x1x2⋯xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤PM 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 x1x2⋯xnN≤P and x1x2⋯xnN≰(P:1M)mP for x1,x2,…,xn∈L, N∈M, and m≥1. Then x1x2⋯xnN≤P and x1x2⋯xnN≰∧m=1∞(P:1M)mP. Since P is ω-absorbing primary, it follows that x1x2⋯xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤PM 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) form=1 and therefore (d)⇒(e) holds.
(3) Suppose thatP∈M is m-almost n-absorbing primary for m≥1 and also suppose that x1x2⋯xnN≤P and x1x2⋯xnN≰∧m=1∞(P:1M)mP for x1,x2,…,xn∈L and N∈M. Then x1x2⋯xmN≤P and x1x2⋯xnN≰(P:1M)mP for some m≥1. Since P is m-almost n-absorbing primary, we have x1x2⋯xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤PM for some i∈{1,2,…,n}. Consequently, P is ω-absorbing primary.
The converse follows from(c)⇒(d).Corollary 9.
LetM be a lattice module over a C-lattice L. If P∈M is ϕ-absorbing primary, where ϕ≤ϕn+2, then P is ω-absorbing primary.Proof.
Suppose thatP∈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)nP≤ϕ(P)≤ϕn+2(P)=(P:1M)n+1P. Consequently, ϕ(P)=(P:1M)kP, for each k≥n. And, by Theorem 8(3), P is ω-absorbing primary.Lemma 10.
LetL be a multiplicative lattice. If p∈L is ϕ-absorbing primary with ϕ(p)=ϕ(p), then p is also ϕ-absorbing.Proof.
(I) Suppose thatx1x2⋯xn+1≤p and x1x2⋯xn+1≰ϕ(p)=ϕ(p), for x1,x2,…,xn+1∈L. If x1x2⋯xi-1xi+1⋯xn+1≰p for i∈{1,2,…,n}, then we have (x1x2⋯xi-1xi+1⋯xn+1)s≰p for any positive integer s and for i∈{1,2,…,n}.
(II) Now, sincex1x2⋯xn+1≤p, there exists a positive integer k such that (x1x2⋯xn+1)k=x1kx2k⋯xn+1k≤p.
(III) Asx1x2⋯xn+1≰ϕ(p), we have x1tx2t⋯xn+1t≰ϕ(p) for a positive integer t.
From the assumption and (I), (II), and (III), it follows that(x1x2⋯xn)k=x1kx2k⋯xnk≤p. Consequently, x1x2⋯xn≤p and therefore p is ϕ-absorbing primary.It is easy to observe that if(L1,∧1,∨1,∘1) and (L2,∧2,∨2,∘2) are multiplicative lattices then L1×L2 is also a multiplicative lattice with componentwise meet, join, and multiplication.Also, ifM1 and M2 are lattice modules over multiplicative lattices L1 and L2, respectively, then M1×M2 is a lattice module over L1×L2 with componentwise meet, join, and multiplication given by (a,b)(N1,N2)=(aN1,bN2), where (a,b)∈L1×L2 and (N1,N2)∈M1×M2.Lemma 11.
LetM=M1×M2 and L=L1×L2, where Mi is a lattice module over C-lattice Li, 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),⇔x1M1≤P1 and y1M2≤P2,⇔(x,y)(1M1,1M2)=(x1M1,y1M2)≤(P1,P2),⇔(x,y)≤(P1,P2):(1M1,1M2).Lemma 12.
LetM=M1×M2 and L=L1×L2, where Mi is a lattice module over C-lattice Li, for i=1,2. Then (P1,P2)M=(P1M1,P2M2), for P1∈M1 and P2∈M2.Proof.
Let(N1,N2)∈M∗ with (N1,N2)≤(P1,P2)M=(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),⇔(a,b)(1M1,1M2)≤(P1M1,P2M2).Consequently,IP1,P2M≤P1M1,P2M2.
Next, let(S1,S2)∈M∗ be such that (S1,S2)≤(P1M1,P2M2): ⇒S1≤P1M1 and S2≤P2M2;⇒S1≤(P1:1M1)1M1 and S2≤(P2:1M2)1M2;⇒S1≤x1M1 and S2≤y1M2 for some x,y∈L with x≤(P1:1M1) and y≤(P2:1M2);⇒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,IIP1M1,P2M2≤P1,P2M.From I and II, result follows.Theorem 13.
LetM=M1×M2 and L=L1×L2, where Mi is a lattice module over C-lattice Li, 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.Proof.
Let(a1,b1),(a2,b2),…,(an,bn)∈L and (N1,N2)∈M be such that (a1,b1)(a2,b2)⋯(an,bn)(N1,N2)=(a1a2⋯a2N1,b1b2⋯b2N2)≤(P1,1M2) and (a1,b1)(a2,b2)⋯(an,bn)(N1,N2)=(a1a2⋯anN1,b1b2⋯bnN2)≰ϕ(P1,1M2).
Since(0,1M2)≤ϕ(P1,1M2), we have (a1,b1)(a2,b2)⋯(an,bn)(N1,N2)=(a1a2⋯anN1,b1b2⋯bnN2)≰(0,1M2) and so 0≠a1a2⋯anN1≤P1. Since P1∈M1 is weakly n-absorbing primary, we have a1a2⋯an≤(P1:1M1) or a1a2⋯ai-1ai+1⋯anN1≤P1M1, for some i∈{1,2,…,n}. This implies that (a1,b1)(a2,b2)⋯(an,bn)=(a1a2⋯an,b1b2⋯bn)≤((P1:1M1),(1M2:1M2))=((P1,1M2):(1M1,1M2)) or (a1,b1)(a2,b2)⋯(ai-1,bi-1)(ai+1,bi+1)⋯(an,bn)(N1,N2)≤(P1M1,1M2)=(P1,1M2)M for some i∈{1,2,…,n}, by Lemmas 11 and 12. Therefore, (P1,1M2) is ϕ-absorbing primary element of M1×M2.Theorem 14.
LetM be a lattice module over a C-lattice L and P∈M. Then P is ϕ-absorbing primary if and only if (P:x1x2⋯xn-1N)=(P:x1x2⋯xn-11M) or (P:x1x2⋯xn-1N)=(ϕ(P):x1x2⋯xn-1N) or (P:x1x2⋯xn-1N)=(P:x1x2⋯xi-1xi+1⋯xn-1N) for some i∈{1,2,…,n-1}, for x1,x2,…,xn-1∈L and N∈M with x1x2⋯xn-1N≰PM.Proof.
Suppose thatP∈M is ϕ-absorbing primary and x1x2⋯xn-1N≰PM, for x1,x2,…,xn-1∈L and N∈M. Let r∈L∗ be such that r≤(P:x1x2⋯xn-1N) which essentially implies that x1x2⋯xn-1rN≤P.
We have the following two cases.
Case 1. If rx1x2⋯xn-1N≰ϕ(P), then as P is ϕ-absorbing primary, we have x1x2⋯xn-1r≤(P:1M) or x1x2⋯xi-1xi+1⋯xn-1rN≤PM for some i∈{1,2,…,n-1}. Therefore, r≤(P:x1x2⋯xn-11M) or r≤(PM:x1x2⋯xi-1xi+1⋯xn-1N) for some i∈{1,2,…,n-1}.
Case 2. If rx1x2⋯xn-1N≤ϕ(P), then r≤(ϕ(P):x1x2⋯xn-1N). So (P:x1x2⋯xn-1N)≤(ϕ(P):x1x2⋯xn-1N).
Now, from Cases 1 and 2, it follows that(P:x1x2⋯xn-1N)=(P:x1x2⋯xn-11M) or (P:x1x2⋯xn-1N)=(ϕ(P):x1x2⋯xn-1N) or (P:x1x2⋯xn-1N)=(PM:x1x2⋯xi-1xi+1⋯xn-1N) for some i∈{1,2,…,n-1}.
Conversely, suppose thatx1x2⋯xnN≤P and x1x2⋯xnN≰ϕ(P), for x1,x2,…,xn∈L and N∈M. If x1x2⋯xn-1N≤PM, then the result is obvious. So, suppose that x1x2⋯xn-1N≰PM.
Now,(P:x1x2⋯xn-1N)=(P:x1x2⋯xn-11M) or (P:x1x2⋯xn-1N)=(ϕ(P):x1x2⋯xn-1N) or (P:x1x2⋯xn-1N)=(PM:x1x2⋯xi-1xi+1⋯xn-1N) for some i∈{1,2,…,n-1}. Since x1x2⋯xnN≤P, we have xn≤(p:x1x2⋯xn-1N). But xn≰(ϕ(P):x1x2⋯xn-1N) and so x1x2⋯xn-1xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤PM for some i∈{1,2,…,n}. Consequently, P is ϕ-absorbing primary.Theorem 15.
LetM=M1×M2×⋯×Mn and L=L1×L2×⋯×Ln, where each Mi is a compactly generated lattice module over a C-lattice Li, 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 ψ-absorbing primary. Then Pi is a ψi-absorbing primary element of Mi, for each i with Pi≠1Mi.Proof.
LetPi≠1Mi, Ni∈Mi, and x1,x2,…,xn∈Li be such that x1x2⋯xnNi≤Pi and x1x2⋯xnNi≰ψi(Pi). Thus (1,…,1,x1,1,…,1)(1,…,1,x2,1,…,1)⋯(1,…,1,xn,1, …,1)(0,…,0,Ni,0,…,0)=(0,…,0,x1x2⋯xnNi,0,…,0)≤P, and (0,…,0,x1x2⋯xnNi,0,…,0)≰ψ(P). As P is ψ-absorbing primary, (1,1,…,x1x2⋯xn,0,…,0)≤(P:1M) or (1,1,…,x1x2⋯xi-1xi+1⋯xnNi,0,…,0)≤PM. Now, by Lemmas 11 and 12, we have x1x2⋯xn≤(Pi:1Mi) or x1x2⋯xi-1xi+1⋯xnNi≤PiM for i∈{1,2,…,n} and consequently, Pi is ψi-absorbing primary element of Mi, for each i.Corollary 16.
LetM=M1×M2×⋯×Mn and L=L1×L2×⋯×Ln, where each Mi is a compactly generated lattice module over a C-lattice Li, 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).Proof.
We haveϕm(P)=(P:1M)m-1P=((P1:1M1)m-1P1,(P2:1M2)m-1P2,…,(Pn:1Mn)m-1Pn)=(ϕm(P1),ϕm(P2),…,ϕm(Pn)) and the result follows from Theorem 15.Theorem 17.
LetM 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 thata1M∈M is m-almost n-absorbing primary and x1x2⋯xnN≤a1M, for x1,x2,…,xn∈L and N∈M. If x1x2⋯xnN≰(a1M:1M)ma1M, then x1x2⋯xn≤(a1M:1M) or x1x2⋯xi-1xi+1⋯xnN≤a1MM for some i∈{1,2,…,n} and therefore a1M is n-absorbing primary.
Now, suppose thatx1x2⋯xnN≤(a1M:1M)ma1M. Since x1x2⋯xnN≤a1M, we have (x1∨a)x2⋯xnN≤a1M.
If(x1∨a)x2⋯xnN≰(a1M:1M)ma1M, then it follows from the fact (x1∨a)x2⋯xnN≤a1M, (x1∨a)x2⋯xnN≰(a1M:1M)ma1M, and a1M is m-almost n-absorbing primary that x1x2⋯xn≤(a1M:1M) or x1x2⋯xi-1xi+1⋯xnN≤a1MM for some i∈{1,2,…,n} and we are done.
So assume that(x1∨a)x2⋯xnN≤(a1M:1M)ma1M. Then as x1x2⋯xnN≤(a1M:1M)na1M, we have ax2⋯xnN≤(a1M:1M)na1M. Next, ax2⋯xnN≤(a1M:1M)na1M and a1M is weak join principal; together they imply that x2⋯xn≤((a1M:1M)na1M:a1M)=(a1M:1M)n∨(0M:a1M)≤a∨(0M:a1M)≤a. Consequently, x2⋯xnN≤aN≤a1M=a1MM which implies that a1M is n-absorbing primary.
The converse follows from Theorem8(2).Note. The results pertaining to ϕ-absorbing elements are essentially the corollaries to the respective results of ϕ-absorbing primary elements, as such the results of ϕ-absorbing elements are the immediate consequences of results of ϕ-absorbing primary elements in this paper.



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