Generalization of primal superideals
Let R be a commutative super-ring with unity 16= 0. A proper super ideal of R is a super ideaI of R such that I 6=R.Letφ:I(R)→I(R)∪ {∅}be any function, where I(R) denotes the set of all proper super ideals of R. A homogeneous element a∈R is φ-prime to Iifra∈I−φ(I) whereris a homogeneous element in R...
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| Опубліковано в: : | Algebra and Discrete Mathematics |
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| Дата: | 2016 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Інститут прикладної математики і механіки НАН України
2016
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Generalization of primal superideals / A. Jaber // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 202-213. — Бібліогр.: 13 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859989150596333568 |
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| author | Jaber, A. |
| author_facet | Jaber, A. |
| citation_txt | Generalization of primal superideals / A. Jaber // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 202-213. — Бібліогр.: 13 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | Let R be a commutative super-ring with unity 16= 0. A proper super ideal of R is a super ideaI of R such that I 6=R.Letφ:I(R)→I(R)∪ {∅}be any function, where I(R) denotes the set of all proper super ideals of R. A homogeneous element a∈R is φ-prime to Iifra∈I−φ(I) whereris a homogeneous element in R, then r∈I. We denote byνφ(I) the set of all homogeneous elements in R that are notφ-prime to I. We define Ito beφ-primal if the set P=([(νφ(I))0+ (νφ(I))1∪ {0}] +φ(I) : ifφ6=φ∅(νφ(I))0+ (νφ(I))1: ifφ=φ∅forms a super ideal of R. For example if we takeφ∅(I) =∅(resp.φ0(I) = 0), aφ-primal superideal is a primal super ideal (resp., a weakly primal super ideal). In this paper we study several generalizations of primal super ideals of R and their properties.
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| first_indexed | 2025-12-07T16:30:31Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 21 (2016). Number 2, pp. 202–213
© Journal “Algebra and Discrete Mathematics”
Generalization of primal superideals
Ameer Jaber
Communicated by Z. Marciniak
Abstract. Let R be a commutative super-ring with unity
1 6= 0. A proper superideal ofR is a superideal I ofR such that I 6= R.
Let φ : I(R) → I(R)∪{∅} be any function, where I(R) denotes the
set of all proper superideals of R. A homogeneous element a ∈ R
is φ-prime to I if ra ∈ I − φ(I) where r is a homogeneous element
in R, then r ∈ I. We denote by νφ(I) the set of all homogeneous
elements in R that are not φ-prime to I. We define I to be φ-primal
if the set
P =
{
[(νφ(I))0 + (νφ(I))1 ∪ {0}] + φ(I) : if φ 6= φ∅
(νφ(I))0 + (νφ(I))1 : if φ = φ∅
forms a superideal of R. For example if we take φ∅(I) = ∅ (resp.
φ0(I) = 0), a φ-primal superideal is a primal superideal (resp., a
weakly primal superideal). In this paper we study several general-
izations of primal superideals of R and their properties.
1. Introduction
A supercase on a ring is a Z2-grading on that ring. In general the
grading on a ring, or a module, usually leads computation by allowing one
to focus on the homogeneous elements, which are simpler and easier than
random elements. However, to do this work you need to know that the
constructions being studied are graded. One approach to this issue is to
2010 MSC: 13A02, 16D25, 16W50.
Key words and phrases: primal superideal, φ-P -primal superideal, φ-prime
superideal.
A. Jaber 203
redefine the constructions entirely in terms of graded modules and avoid
any consideration of non-graded modules or non-homogeneous elements.
Unfortunately, while such an approach helps to understand the graded
modules, it will only help to understand the original construction, where
the graded version of the concept coincide with original one. Therefore,
notably, the studying of the graded rings (or modules) is very important.
Because of the importance of the grading, the author made many
researches in different subjects in mathematics in super-rings and graded
rings few years ago. For example in [1,2,4], the author studied existence of
superinvolutions and pseudo superinvolutions of kinds one and two, also
in [3, 5] he studied Division Z3-Algebra, and primitive Z3-algebra with
Z3-involution. Moreover, in [7] he studied ∆-supergraded submodules and
in [6] he studied product of graded submodules. Finally, in [8] the author
studied weakly primal graded superideals.
A few years ago Y. A. Bahturin and A. Giambruno in [12] studied
Group Gradings on associative algebras with involution.
Let R be any ring with unity, then R is called a super-ring if R is
a Z2-graded ring such that if a, b ∈ Z2 then RaRb ⊆ Ra+b where the
subscripts are taken modulo 2. Let h(R) = R0 ∪R1. Then h(R) is the set
of homogeneous elements in R and 1 ∈ R0.
Throughout, R will be a commutative super-ring with unity. By a
proper superideal of R we mean a superideal I of R such that I 6= R. We
will denote the set of all proper superideals of R by I(R). If I and J are
in I(R), then the superideal {r ∈ R : rJ ⊆ I} is denoted by (I : J). Let
φ : I(R) → I(R) ∪ {∅} be any function and let I ∈ I(R) , we say that I
is a φ-prime if whenever x, y ∈ h(R) with xy ∈ I − φ(I), then x ∈ I or
y ∈ I. Since I − φ(I) = I − (φ(I) ∩ I), there is no loss of generality to
assume that φ(I) ⊆ I for every proper superideal I of R.
Given two functions ψ1, ψ2 : I(R) → I(R) ∪ {∅}, we define ψ1 6 ψ2
if ψ1(I) ⊆ ψ2(I) for each I ∈ I(R).
Let φ : I(R) → I(R) ∪ {∅} be any function, then an element a ∈ h(R)
is φ-prime to I, if whenever ra ∈ I − φ(I), where r ∈ h(R), then r ∈ I.
That is a ∈ h(R) is φ-prime to I, if
h((I : a)) − h((φ(I) : a)) ⊆ h(I).
Let νφ(I) be the set of all homogeneous elements in R that are not φ-prime
to I. We define I to be φ-primal if the set
P =
{
[(νφ(I))0 + (νφ(I))1 ∪ {0}] + φ(I) : if φ 6= φ∅
(νφ(I))0 + (νφ(I))1 : if φ = φ∅
204 Generalization of primal superideals
forms a superideal in R. In this case we say that I is a φ-P -primal
superideal of R, and P is the adjoint superideal of I.
In the next example we give some famous functions φ : I(R) →
I(R) ∪ {∅} and their corresponding φ-primal superideals.
Example 1.1.
• φ∅, φ∅(I) = ∅∀I ∈ I(R) — primal superideal.
• φ0, φ0(I) = {0}∀I ∈ I(R) — weakly primal superideal.
• φ2, φ2(I) = I2∀I ∈ I(R) — almost primal superideal.
• φn, φn(I) = In∀I ∈ I(R) — n-almost primal superideal.
• φω, φω(I) = ∩∞
n=1I
n∀I ∈ I(R) — ω-primal superideal.
Observe that φ∅ 6 φ0 6 φω 6 · · · 6 φn+1 6 φn 6 · · · 6 φ2.
For the nongraded case one can easily check that if I is a φ-P -primal
ideal of R, with φ 6= φ∅, then P = (νφ(I) ∪ {0}) + φ(I) if and only if
P = νφ(I) ∪ φ(I). But if φ = φ∅ then P = νφ(I).
Y. Darani in [13] defined that for a commutative ring R with unity
and for a function φ : I(R) → I(R) ∪ {∅} a proper ideal I of R is a
φ-P-primal ideal of R if P = φ(I) ∪ νφ(I) is an ideal in R, where νφ(I) is
the set of all elements in R that are not φ-prime to I.
By comparing the two definitions (in the trivial case and in the
supercase), we can see that the definition of φ-primal superideals is a
generalization of the definition of the φ-primal ideals to the supercase.
In section 2, we give some examples and properties of φ-primal su-
perideals of R. Also, we prove that if R is φ-torsion free super-ring, then
every φ-primary superideal of R is φ-primal and hence if R is torsion free
super-ring then every weakly primary (i.e., φ0-primary) superideal of R
is weakly primal.
In section 3, we introduce some conditions under which φ-primal
superideals are primal.
2. φ-Primal superideals
Let R be a commutative super-ring with unity 1 6= 0 ∈ R0. Let
φ : I(R) → I(R) ∪ {∅} be any function and let I be a proper superideal
of R. Suppose that νφ(I) is the set of all homogeneous elements in R that
are not φ-prime to I, we recall that I is a φ-primal superideal of R if the
set
P =
{
[(νφ(I))0 + (νφ(I))1 ∪ {0}] + φ(I) : if φ 6= φ∅
(νφ(I))0 + (νφ(I))1 : if φ = φ∅
A. Jaber 205
forms a superideal in R. In this case P is called the adjoint superideal
of I.
In the next examples we show that the concepts "primal superideals"
and "φ-primal superideals" are different.
Example 2.1. Let R = Z24 + uZ24, where u2 = 0, be a commutative
super-ring and assume that φ = φ0. Let I = 8Z24 + uZ24.
(1) Since 0 6= 2̄ · 4̄ ∈ I with 2̄, 4̄ 6∈ I, then we get that 2̄ and 4̄ are not
φ-prime to I. Easy computations imply that 2̄ + 4̄ = 6̄ is φ-prime to I.
Thus we obtain that I is not a φ-primal superideal of R.
(2) Set P = 2Z24 + uZ24. We show that I is a primal superideal
of R. It is easy to check that every element of h(P ) is not prime to I.
Conversely, assume that ā ∈ h(R)−h(P ), then ā ∈ Z24 with gcd(a, 8) = 1.
If ā · n̄ ∈ I for some n̄ ∈ Z24, then 8 divides n; hence n̄ ∈ I. Therefore,
h(P ) is exactly the set of elements in h(R) which are not prime to I.
Thus I is a primal superideal of R.
Example 2.2. Let φ = φ0, and let T (R) be the collection of all ho-
mogeneous zero divisors of R. If R is not a superdomain such that
Z(R) = T0(R)+T1(R) is not a superideal of R, then the trivial superideal
of R is a φ-primal superideal which is not primal.
According to Examples 2.1 and 2.2 a primal superideal of R need
not to be φ-primal and a φ-primal superideal of R need not to be primal.
In the next lemma we show that if I is a φ-primal superideal in R,
then I ⊆ P . The same result for the non graded case has been proved
in [13].
Lemma 2.3. Let I be a superideal of R, and let φ : I(R) → I(R)∪{∅} be
any function. Suppose that I is φ-primal superideal of R with the adjoint
superideal P . Then
(1) I ⊆ P .
(2) h(P ) = h(φ(I)) ∪ νφ(I).
Proof. (1) Let r be any homogeneous element in I, if r ∈ φ(I), then r ∈ P .
If r ∈ h(I) − h(φ(I)), then 1.r ∈ I − φ(I) with 1 6∈ I, hence r ∈ P . Thus,
I ⊆ P .
(2) It is trivial that νφ(I) ⊆ h(P ) − h(φ(I)). For the reverse inclusion,
let x ∈ h(P ) − h(φ(I)) then x = xα + yα, where xα 6= 0 ∈ νφ(I) and
yα ∈ (φ(I))α, for some α in Z2. Since xα 6= 0 ∈ νφ(I), there exists
r ∈ h(R) − h(I) with rxα ∈ I − φ(I). Thus, rx = rxα + ryα ∈ I − φ(I)
since ryα ∈ φ(I). Hence x ∈ νφ(I).
206 Generalization of primal superideals
Proposition 2.4. Let I, P be proper superideals of R. Then the following
statements are equivalent.
(1) I is a φ-primal superideal of R with the adjoint superideal P .
(2) For x ∈ h(R) with x 6∈ h(P ) −h(φ(I)) we have h((I : x)) = h(I) ∪
h((φ(I) : x)). If x ∈ h(P ) −h(φ(I)) then h((I : x)) % h(I) ∪h((φ(I) : x)).
Proof. (1) ⇒ (2) If x ∈ h(P ) − h(φ(I)), then x ∈ νφ(I), so there exists
r ∈ h(R) − h(I) with rx ∈ I − φ(I). Thus r ∈ h((I : x)) and r 6∈ h(I) ∪
h((φ(I) : x)). Since it is easy to see that h((I : x)) ⊇ h(I) ∪ h((φ(I) : x)),
we have that h((I : x)) % h(I) ∪ h((φ(I) : x)).
Now let x 6∈ h(P ) − h(φ(I)), where x ∈ h(R), then x 6∈ νφ(I) hence x
is φ-prime to I. Let r ∈ h((I : x)), if rx 6∈ φ(I) then r ∈ h(I). If rx ∈ φ(I)
then r ∈ h((φ(I) : x)). Hence
h((I : x)) ⊆ h(I) ∪ h((φ(I) : x)) ⊆ h((I : x)).
(2) ⇒ (1) From part (2) we have h(P ) − h(φ(I)) = νφ(I). Thus I is a
φ-primal superideal of R.
Theorem 2.5. If I is a φ-primal superideal of R, then
P =
{
[(νφ(I))0 + (νφ(I))1 ∪ {0}] + φ(I) : if φ 6= φ∅
(νφ(I))0 + (νφ(I))1 : if φ = φ∅
is a φ-prime superideal of R.
Proof. Suppose that a, b ∈ h(R) − h(P ) we show that ab ∈ φ(P ) or
ab 6∈ P . Assume that ab 6∈ φ(P ), then ab 6∈ φ(I), since φ(I) ⊆ φ(P ).
Let rab ∈ I − φ(I) for some r ∈ h(R). Then by Proposition 2.4, we
have ra ∈ h((I : b)) = h(I) ∪ h((φ(I) : b)), but ra 6∈ (φ(I) : b); hence
ra ∈ h(I). Moreover ra 6∈ h(φ(I)), for if ra ∈ h(φ(I)), then rab ∈ h(φ(I)),
which is a contradiction. Therefore, ra ∈ h(I) − h(φ(I)) and again by
Proposition 2.4, r ∈ h((I : a)) = h(I) ∪ h((φ(I) : a)). Since ra 6∈ φ(I),
we have r 6∈ h((φ(I) : a)), so r ∈ h(I). Hence ab is φ-prime to I which
implies that ab 6∈ P .
Remark 2.6. Let I is a φ-primal superideal of R then by Theorem 2.5,
P =
{
[(νφ(I))0 + (νφ(I))1 ∪ {0}] + φ(I) : if φ 6= φ∅
(νφ(I))0 + (νφ(I))1 : if φ = φ∅
is a φ-prime superideal of R. In this case P is called the φ-prime adjoint
superideal (simply adjoint superideal) of I, and I is called a φ-P -primal
superideal of R.
A. Jaber 207
The next result shows that every φ-prime superideal of R is φ-primal.
Theorem 2.7. Every φ-prime superideal of R is φ-primal.
Proof. Let P be a φ-prime superideal ofR, we show that P is a φ-P -primal
superideal of R. Thus we must prove that
P =
{
[(νφ(P ))0 + (νφ(P ))1 ∪ {0}] + φ(P ) : if φ 6= φ∅
(νφ(P ))0 + (νφ(P ))1 : if φ = φ∅
Case 1. Suppose that P 6= φ(P ). We show that h(P ) − h(φ(P )) = νφ(P ).
Let a ∈ h(P ) − h(φ(P )). Then a.1 ∈ P − φ(P ) with 1 6∈ P , so a ∈ νφ(P ).
On the other hand let a 6∈ h(P )−h(φ(P )). If a ∈ h(φ(P )), then ra ∈ φ(P )
for all r ∈ h(R), so a is φ-prime to P and hence a 6∈ νφ(P ). If a 6∈ h(φ(P )),
then a 6∈ P , so for any rα ∈ Rα with rαa ∈ P − φ(P ) we have rα ∈ Pα,
since P is φ-prime. Thus a is φ-prime to P , hence a 6∈ νφ(P ). Therefore,
h(P ) − h(φ(P )) = νφ(P ) which implies that P is a φ-P -primal superideal
of R.
Case 2. Suppose that P = φ(P ) then it is easy to check that νφ(P ) = ∅,
hence P is a φ-P -primal superideal of R.
In the next example we introduce a φ-P -primal superideal I of R such
that I itself is not φ-prime.
Example 2.8. Let φ = φ0 and let R = Z8 + uZ8 where u2 = 0. Then R
is a commutative super-ring with unity. If I = 4Z8 + uZ8, then I is not a
φ-prime superideal of R, since 2̄ · 2̄ 6= 0 ∈ I, but 2̄ 6∈ I. Let P = 2Z8 +uZ8,
we show that I is a φ-P -primal superideal of R. It is enough to show that
ν(I) = h(P ) − {0}. Let 0 6= ā ∈ h(P ), if ā ∈ 2Z8 then ā = 2k ∈ Z8. If k
is an odd number, then 0 6= 2̄ā ∈ I, but 2̄ 6∈ I, and if k is an even number
0 6= 1̄ā ∈ I with 1̄ 6∈ I; hence ā ∈ ν(I). If ā ∈ uZ8 then ā ∈ I ⊆ ν(I). On
the other hand, if ā ∈ h(R) − h(P ), then ā is an odd number in Z8. If
0 6= ām̄ ∈ I for some m̄ ∈ Z8 then 4 divides am and so, 4 divides m since
(4, a) = 1; hence m̄ ∈ I. Thus I is a φ-P -primal superideal of R.
Let φ : I(R) → I(R) ∪ {∅} be any function. We assume that for
any I, J ∈ I(R), φ(J) ⊆ φ(I) if J ⊆ I. We produced in Example 2.2 a
ψ2-primal which is not ψ1-primal, where ψ1 6 ψ2. In the next theorem
we give the condition on ψ2-P -primal superideal to be ψ1-P -primal.
Theorem 2.9. Suppose that ψ1 6 ψ2, where ψ1 and ψ2 are maps from
I(R) into I(R) ∪ {∅}, and let I be a ψ2-P -primal superideal of R, with
I0Iα 6= ψ2(I)α for all α ∈ Z2. If P is a prime superideal of R, then I is
ψ1-P -primal.
208 Generalization of primal superideals
Proof. Since I is a ψ2-P -primal superideal of R, then
P =
{
[(νψ2
(I))0 + (νψ2
(I))1 ∪ {0}] + ψ2(I) : if ψ2 6= φ∅
(νψ2
(I))0 + (νψ2
(I))1 : if ψ2 = φ∅
To show that I is a ψ1-P -primal superideal of R we must prove that
P =
{
[(νψ1
(I))0 + (νψ1
(I))1 ∪ {0}] + ψ1(I) : if ψ1 6= φ∅
(νψ1
(I))0 + (νψ1
(I))1 : if ψ1 = φ∅
If ψ2 = φ∅, then ψ1 = ψ2 and hence we have that P = νψ1
(I))0 +(νψ1
(I))1
which implies that I is a ψ1-P -primal superideal of R. Now we may assume
that ψ2 6= φ∅, so we need to prove that
P =
{
[(νψ1
(I))0 + (νψ1
(I))1 ∪ {0}] + ψ1(I) : if ψ1 6= φ∅
(νψ1
(I))0 + (νψ1
(I))1 : if ψ1 = φ∅
.
Let a ∈ νψ2
(I), then there exists r ∈ h(R) − h(I) with rs ∈ I − ψ2(I) ⊆
I − ψ1(I), so a ∈ νψ1
(I) which implies that
(νψ2
(I))0 + (νψ2
(I))1 ⊆ (νψ1
(I))0 + (νψ1
(I))1 (1)
Now, let a ∈ h(ψ2(I)), if a 6∈ ψ1(I) then a ∈ I −ψ1(I), so 1.a ∈ I −ψ1(I)
with a 6∈ I, hence a ∈ νψ1
(I). Therefore,
ψ2(I) ⊆
{
[(νψ1
(I))0 + (νψ1
(I))1 ∪ {0}] + ψ1(I) : if ψ1 6= φ∅
(νψ1
(I))0 + (νψ1
(I))1 : if ψ1 = φ∅
. (2)
From (1) and (2) we have that
P ⊆
{
[(νψ1
(I))0 + (νψ1
(I))1 ∪ {0}] + ψ1(I) : if ψ1 6= φ∅
(νψ1
(I))0 + (νψ1
(I))1 : if ψ1 = φ∅
. (3)
Since ψ1(I) ⊆ ψ2(I) ⊆ P , by (3)
P =
{
[(νψ1
(I))0 + (νψ1
(I))1 ∪ {0}] + ψ1(I) : if ψ1 6= φ∅
(νψ1
(I))0 + (νψ1
(I))1 : if ψ1 = φ∅
if νψ1
(I) ⊆ P .
Let a ∈ (νψ1
(I))α. Then there exists rβ ∈ Rβ−Iβ with arβ ∈ I−ψ1(I).
If arβ ∈ I − ψ2(I), then a ∈ νψ2
(I) ⊆ P . So we may assume that arβ 6∈
A. Jaber 209
I − ψ2(I), hence arβ ∈ ψ2(I). First suppose that aIβ 6⊆ (ψ2(I))αβ, say
asβ ∈ Iαβ − (ψ2(I))αβ with sβ ∈ Iβ . Then a(rβ + sβ) = arβ + asβ 6∈ ψ2(I)
with rβ + sβ ∈ Rβ − Iβ , hence a ∈ νψ2
(I) ⊆ P . Therefore, we may assume
that aIβ ⊆ (ψ2(I))αβ .
Now suppose that rβI0 6⊆ (ψ2(I))β, then there exists c ∈ I0 with
rβc ∈ Iβ−(ψ2(I))β . Since a2 ∈ R0, we have that (a2+c)rβ ∈ Iβ−(ψ2(I))β
with rβ 6∈ Iβ, so a2 + c ∈ P0, but c ∈ I0 ⊆ P0, therefore a2 ∈ P and
hence a ∈ P , since P is a prime superideal. So we may assume that
rβI0 ⊆ (ψ2(I))β. Since (I0Iβ) 6= (ψ2(I))β there exists a ∈ I0 and b ∈ Iβ
with ab 6∈ (ψ2(I))β . Thus, (a2 +a)(rβ +b) = a2rβ +a2b+arβ +ab 6∈ ψ2(I),
so (a2 + a)(rβ + b) ∈ I − ψ2(I) with rβ + b ∈ Rβ − Iβ which implies that
a2 + a ∈ (νψ2
(I))0 ⊆ P0, hence a2 ∈ P0 ⊆ P and then a ∈ P , since P is a
prime superideal of R. Therefore, νψ1
(I) ⊆ P , so
P =
{
[(νψ1
(I))0 + (νψ1
(I))1 ∪ {0}] + ψ1(I) : if ψ1 6= φ∅
(νψ1
(I))0 + (νψ1
(I))1 : if ψ1 = φ∅
and hence I is a ψ1-P -primal superideal of R.
We end the section by proving the following results about the relation-
ship between φ-primary and φ-primal superideals. For more properties
about primary and primal superideals see [8, section 4].
Definition 2.10. Let φ : I(R) → I(R) ∪ {∅} be any function such that
φ 6= φ∅, then R is a φ-torsion free if ab ∈ φ(P ) where P ∈ I(R), then
a ∈ φ(P ) or b ∈ φ(P ).
For example if φ = φ0, then a φ-torsion free super-ring is just a torsion
free super-ring.
Theorem 2.11. Let φ : I(R) → I(R) ∪ {∅} be any function, where
φ 6= φ∅, and let R be a φ-torsion free. Then every φ-primary superideal
of R is φ-primal.
Proof. Let I be a φ-primary superideal of R. We show that
√
I = [(νφ(I))0 + (νφ(I))1 ∪ {0}] + φ(I).
(⊇) Let r ∈ νφ(I), then there exists a ∈ h(R) − h(I) with ra ∈ I − φ(I)
which implies that r ∈
√
I, since I is φ-primary. Moreover, φ(I) ⊆ I ⊆
√
I.
(⊆) Let b ∈ h(
√
I). If b ∈ φ(I), then done. So, we may assume that
b 6∈ φ(I). Let n be the smallest positive integer such that bn ∈ I. Suppose
210 Generalization of primal superideals
n = 1. If b ∈ φ(I), then done. If b 6∈ φ(I), then 1.b ∈ I − φ(I) and 1 6∈ I
so b ∈ νφ(I). Therefore we may assume that n > 1. If bn ∈ φ(I), then
bn = bn−1b ∈ φ(I) and bn−1 6∈ φ(I), since bn−1 6∈ I and φ(I) ⊆ I, which
is a contradiction since R is φ-torsion free. So, bn = bn−1b ∈ I − φ(I) and
bn−1 6∈ I, hence b ∈ νφ(I).
Corollary 2.12. If R is a torsion free, then every weakly primary su-
perideal of R is weakly primal.
3. Conditions on φ-primal superideals
In this section, we introduce some conditions under which φ-primal
superideals are primal.
Let φ : I(R) → I(R) ∪ {∅} be any function. We have to remind that
if I is a φ-P -primal superideal of R, then
P =
{
[(νφ(I))0 + (νφ(I))1 ∪ {0}] + φ(I) : if φ 6= φ∅
(νφ(I))0 + (νφ(I))1 : if φ = φ∅
is a φ-prime superideal of R.
Definition 3.1. Let r be a homogeneous element in R, then |r| = α if
r ∈ Rα for some α ∈ Z2.
In the next theorem we provide some conditions under which a φ-
primal superideal is primal.
Theorem 3.2. Let R be a commutative super-ring with unity and let
φ : I(R) → I(R) ∪ {∅} be any function. Suppose that I is a φ-P -primal
superideal of R with IγIδ * φ(I) for each γ, δ ∈ Z2. If P is a prime
superideal of R, then I is P -primal.
Proof. Assume that a is a homogeneous element in P . Then a ∈ φ(I)
or a ∈ (νφ(I))α for some α ∈ Z2 or a = bβ + cβ where bβ ∈ (νφ(I))β
and cβ ∈ φ(I) for some β ∈ Z2. If the first two cases hold, then a is
not prime to I, since it is not φ-prime to I. In the last case, let d be a
homogeneous element in R such that d 6∈ I with bβd ∈ I − φ(I). Then
ad = bβd + cβd ∈ I − φ(I), because ad ∈ φ(I) implies that bβd ∈ φ(I),
since cβd ∈ φ(I) which is a contradiction. Thus a is not φ-prime to I and
hence a is not prime to I. Now assume that b ∈ h(R) is not prime to I,
so rb ∈ I for some homogeneous element r ∈ R− I. If rb 6∈ φ(I), then b
is not φ-prime to I, so b ∈ P . Thus assume that rb ∈ φ(I). Suppose that
A. Jaber 211
|r| = α. First suppose that bIα * φ(I). Then, there exists r′ ∈ Iα such
that br′ 6∈ φ(I). So b(r + r′) ∈ I − φ(I), where r + r′ is a homogeneous
element in R − I, implies that b is not φ-prime to I, that is b ∈ P .
Therefore, we may assume that bIα ⊆ φ(I). Let |b| = β. If rIβ * φ(I),
then rc 6∈ φ(I) for some c ∈ Iβ. In this case r(b + c) ∈ I − φ(I) with
r ∈ R − I, that is b + c ∈ P and hence b ∈ P , since c ∈ I ⊆ P . So we
may assume that rIβ ⊆ φ(I). Since IαIβ * φ(I), there are b′ ∈ Iα and
a′ ∈ Iβ with b′a′ 6∈ φ(I). Then (b+ a′)(r + b′) ∈ I − φ(I), where r + b′ is
a homogeneous element in R − I, implies that b+ a′ is a homogeneous
element in P . On the other hand a′ ∈ I ⊆ P , so that b ∈ P . We have
already shown that P is exactly the set of all elements of R that are not
prime to I. Hence I is P -primal.
Let R and S be commutative super-rings. It is easy to prove that the
prime superideals of R× S have the forms P × S or R×Q where P is a
prime superideal of R and Q is a prime superideal of S. Also we have the
following two propositions about primal superideals of R× S. We leave
the easy proof for the next two results to the reader. For the trivial case
(i.e., R1 = {0}) they have proved in [10, Lemma 13] and [9, Theorem 16].
Proposition 3.3. Let R and S be commutative super-rings. If P is a
primal superideal of R and Q is a primal superideal of S, then P ×S and
R×Q are primal superideals of R× S.
Proposition 3.4. Let R1 and R2 be commutative super-rings with unities
and let ψi : I(R) → I(R) ∪ {∅} be functions. Let φ = ψ1 × ψ2. Then
φ-primes of R1 ×R2 have exactly one of the following three types:
(1) I1 × I2 where Ii is a proper superideal of Ri with ψi(Ii) = Ii;
(2) I1 × R2 where I1 is a ψ1-prime of R1 which must be prime if
ψ2(R2) 6= R2;
(3) R1 × I2 where I2 is a ψ2-prime of R2 which must be prime if
ψ1(R1) 6= R1.
Now let R1, R2 be commutative super-rings with unities and let
R = R1 × R2. Let φ : I(R) → I(R) ∪ {∅} be a function. In the next
theorem, we provide some conditions under which a φ-primal superideal
of R is primal, but first we start with the following remark.
Remark 3.5. Let I be a proper superideal of a commutative super-ring
R and let φ : I(R) → I(R)∪{∅} be a function. If a homogeneous element
a is not φ-prime to I, then there is a homogeneous element r in R − I
such that ar ∈ I − φ(I) ⊆ I so a is not prime to I.
212 Generalization of primal superideals
Theorem 3.6. Let R1, R2 be commutative super-rings with unities and let
R = R1 ×R2. Let ψi : I(R) → I(R) ∪ {∅} be functions with ψi(Ri) 6= Ri
for i = 1, 2. Let φ = ψ1 × ψ2. Assume that P is a superideal of R with
φ(P ) 6= P . If I is a φ-P -primal superideal of R, then either I = φ(I) or
I is primal.
Proof. Suppose φ(I) 6= I. By Theorem 2.5, P is a φ-prime superideal of R.
Therefore, by Proposition 3.4, P has one of the following three cases.
Case 1. P = P1 ×P2 where Pi is a proper superideal of Ri with ψi(Pi) = Pi
for i = 1, 2. In this case φ(P ) = P , a contradiction.
Case 2. P = P1 × R2 where P1 is a ψ1-prime superideal of R1. Since
ψ2(R2) 6= R2, by Proposition 3.4(2), P1 is a prime superideal of R1 and
so P is a prime superideal of R.
We will show that I2 = R2. Since I 6= φ(I), there exists a homogeneous
element (a, b) in I − φ(I). So (a, 1)(1, b) = (a, b) ∈ I − φ(I). If (a, 1) 6∈ I,
then (1, b) is not φ-prime to I, hence (1, b) ∈ P = P1 × P2, so 1 ∈ P1 a
contradiction. Thus (a, 1) ∈ I = I1 × I2 i.e., 1 ∈ I2 that is I2 = R2.
Now we prove that I1 is a P1-primal superideal of R1. Let a1 be a
homogeneous element in P1. Then (a1, 0) ∈ P1 × R2 = P . If (a1, 0) ∈
φ(I) = ψ1(I1) × ψ2(R2), then a1 ∈ ψ1(I1) ⊆ I1 so a1 is not prime to I1.
Therefore, we may assume that (a1, 0) ∈ νφ(I). In this case there exists a
homogeneous element (r1, r2) ∈ R− I such that (a1, 0)(r1, r2) ∈ I − φ(I)
so a1r1 ∈ I1 − ψ1(I1) with r1 ∈ R1 − I1, since R − I = (R1 − I1) × R2,
implies that a1 is not ψ1-prime to I1, hence by Remark 3.5, a1 is not
prime to I1. Conversely, let b1 be a homogeneous element in R1 such that
b1 is not prime to I1. Then there exists a homogeneous element c1 in
R1 − I1 with b1c1 ∈ I1. Since ψ2(R2) 6= R2, (b1, 1)(c1, 1) = (b1c1, 1) ∈
I1 ×R2 − (I1 × ψ2(R2)) ⊆ I − φ(I) with (c1, 1) ∈ R− I. Hence (b1, 1) is
not φ-prime to I which implies that (b1, 1) ∈ P = P1 ×R2 and so b1 ∈ P1.
We have already shown that the set of homogeneous elements in P1
consists exactly of the homogeneous elements of R1 that are not prime
to I1. Hence I1 is P1-primal superideal of R1 so by Proposition 3.3, I is a
P -primal superideal of R.
Case 3. P = R1 ×P2 where P2 is a ψ2-primal superideal of R2. The proof
of case(3) is similar to that of case(2).
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Contact information
A. Jaber Department of Mathematics, The Hashemite
University, Zarqa 13115, Jordan
E-Mail(s): ameerj@hu.edu.jo
Received by the editors: 21.09.2015
and in final form 14.02.2016.
|
| id | nasplib_isofts_kiev_ua-123456789-155239 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T16:30:31Z |
| publishDate | 2016 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Jaber, A. 2019-06-16T14:32:29Z 2019-06-16T14:32:29Z 2016 Generalization of primal superideals / A. Jaber // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 202-213. — Бібліогр.: 13 назв. — англ. 1726-3255 2010 MSC:13A02, 16D25, 16W50. https://nasplib.isofts.kiev.ua/handle/123456789/155239 Let R be a commutative super-ring with unity 16= 0. A proper super ideal of R is a super ideaI of R such that I 6=R.Letφ:I(R)→I(R)∪ {∅}be any function, where I(R) denotes the set of all proper super ideals of R. A homogeneous element a∈R is φ-prime to Iifra∈I−φ(I) whereris a homogeneous element in R, then r∈I. We denote byνφ(I) the set of all homogeneous elements in R that are notφ-prime to I. We define Ito beφ-primal if the set P=([(νφ(I))0+ (νφ(I))1∪ {0}] +φ(I) : ifφ6=φ∅(νφ(I))0+ (νφ(I))1: ifφ=φ∅forms a super ideal of R. For example if we takeφ∅(I) =∅(resp.φ0(I) = 0), aφ-primal superideal is a primal super ideal (resp., a weakly primal super ideal). In this paper we study several generalizations of primal super ideals of R and their properties. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Generalization of primal superideals Article published earlier |
| spellingShingle | Generalization of primal superideals Jaber, A. |
| title | Generalization of primal superideals |
| title_full | Generalization of primal superideals |
| title_fullStr | Generalization of primal superideals |
| title_full_unstemmed | Generalization of primal superideals |
| title_short | Generalization of primal superideals |
| title_sort | generalization of primal superideals |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155239 |
| work_keys_str_mv | AT jabera generalizationofprimalsuperideals |