# δ-Primary Hyperideals on Commutative Hyperrings

Authors: Elif Ozel Ay; Gürsel Yesilot; Deniz Sonmez
Journal: International Journal of Mathematics and Mathematical Sciences
 (2017)

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## Abstract

The purpose of this paper is to define the hyperideal expansion. Hyperideal expansion is associated with prime hyperideals and primary hyperideals. Then, we define some of their properties. Prime and primary hyperideals’ numerous results can be extended into expansions.


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## Body

## 1. Introduction

The hyperstructure theory was introduced by Marty (1934). Hyperstructures have many applications to several sectors of both pure and applied mathematics. A hypergroup in the sense of Marty is a nonempty setH endowed by a hyperoperation ∗:H×H→P∗H [1], the set of the entire nonempty set H, which satisfies the associative law and reproduction axiom. Canonical hypergroups are a special class of the hypergroup of Marty. The more general structure that satisfies the ring-like axioms is the hyperring in the general sense: (R,+,·) is a hyperring if + and · are two hyperoperations such that (R,+) is a hypergroup and · is an associative hyperoperation, which is distributive with respect to +. There are different notions of hyperrings [1]. If only the addition + is hyperoperation and the multiplication · is usual operation, then we say that R is an additive hyperring. A special case of this type is the hyperring introduced by Krasner (1957) [2]. Also, Krasner (1983) introduced a class of hyperring and hyperfields and the quotient hyperrings and hyperfields. If only · is a hyperoperation, we shall say that R is a multiplicative hyperring [2]. Rota (1982) introduced the multiplicative hyperrings; subsequently, many authors worked on this field (Nakassis, 1988; Olson and Ward, 1997; Procesi and Rota, 1999; Rota, 1996) [2]. Algebraic hyperstructures have been studied in the following decades and nowadays by many mathematicians.Although theδ-primary ideals have been investigated by Dongsheng [3], the concept of δ-primary hyperideals which unify prime hyperideals and primary hyperideals has not been studied yet. So, this work shows some elementary properties of the hyperideal expansion; then we show some new results of δ-primary hyperideals. After this introductory section, Section 2 is devoted to some definitions and properties related to δ primary ideals and hyperideals that will be needed later. In Section 3, the definitions of hyperideal expansion and δ primary hyperideals will be given and some basic properties of these concepts will be studied.


## 2. Preliminaries

Throughout this paper(R,+,·) denotes the Krasner hyperring.Definition 1 (see [4]).
A Krasner hyperring is an algebraic structure(R,+,·) which satisfies the following axioms:(1)
(R,+) is a canonical hypergroup; that is,
(i)
for everyx,y,z∈R, x+(y+z)=(x+y)+z,(ii)
for everyx,y∈R, x+y=y+x,(iii)
there exists0∈R such that 0+x={x} for every x∈R,(iv)
for everyx∈R there exists a unique element x′∈R such that 0∈x+x′,(v)
z∈x+y implies y∈-x+z and x∈z-y,(2)
(R,·) is a semigroup having zero as a bilaterally absorbing element; that is, x·0=0·x=0.(3)
The multiplication is distributive with respect to the hyperoperation +.Definition 2 (see [2]).
Let(R,+,·) be a hyperring and A be a nonempty subset of R. Then A is said to be a subhyperring of R if (A,+,·) is itself a hyperring.Definition 3 (see [1]).
A subhyperringA of a hyperring R is a left (right) hyperideal of R if ra∈A (ar∈A) for all r∈R and a∈A. A is called a hyperideal if A is both a left and a right hyperideal.Lemma 4 (see [2]).
A nonempty subsetA of a hyperring R is a hyperideal if and only if(1)
a,b∈A implies a-b⊆A,(2)
a∈A and r∈R imply ra∈A.Definition 5 (see [2]).
LetR1 and R2 be hyperrings. A mapping φ from R1 into R2 is said to be a good (strong) homomorphism if, for all a,b∈R1,(1)φa+b=φa+φb,φa·b=φa·φb,φ0=0.Definition 6 (see [1]).
Letf:R→S be a hyperring homomorphism. The kernel of f, denoted ker(f), is the set of elements of R that map to 0 in S; that is, ker(f)=x∈R∣fx=0.Definition 7 (see [2]).
A hyperidealP of a hyperring R is called a prime hyperideal if whenever a·b∈P, either a∈P or b∈P.Definition 8 (see [2]).
LetI be a hyperideal of the hyperring R. Then the radical of I, denoted by I, is defined as I={x∣xn∈I for some n∈N}.Definition 9 (see [2]).
A hyperidealI of a hyperring R is called a primary hyperideal if whenever a·b∈P, either a∈P or b∈P.


## 3. Hyperideal Expansion andδ Primary Hyperideals

Definition 10.
An expansion of hyperideals, or briefly hyperideal expansion, is a functionδ which assigns to each hyperideal I of a hyperring R another hyperideal δ(I) of the same ring such that the following conditions are satisfied: (i)
I⊆δ(I).(ii)
P⊆Q implies δ(P)⊆δ(Q) for P,Q hyperideals of R.Example 11.
LetId(R) denote the set of all hyperideals of the hyperring R. The identity function δ0, where δ(I)=I for every I∈Id(R), is an expansion of hyperideals.
For eachI hyperideal define δ1(I)=I, the radical of I. Then δ1 is an expansion of hyperideals.Definition 12.
Given an expansionδ of hyperideals, a hyperideal I of R is called δ-primary if ab∈I and a∉I imply b∈δ(I) for all a,b∈R.
Obviously the definition ofδ-primary hyperideals can be also stated as ab∈I and a∉δ(I) implies b∈I for all a,b∈R.Example 13.
(1) A Hyperideal
I
Is
δ
0-Primary If and Only If It Is Prime. Let I be δ0-primary hyperideal. We show that I is prime. Assume that ab∈I and that I is δ0-primary a∈I or b∈δ0(I)=I so I is a prime hyperideal.
Conversely, letI be a prime hyperideal. Assume that ab∈I. Since I is prime a∈I or b∈I=δ0(I) and I is δ0-primary.
(2) A Hyperideal
I
is
δ
1-Primary If and Only If It Is Primary. Let I be δ1-primary. We show that I is primary. Assume that ab∈I. Since I is δ1-primary then we can say that a∈I or b∈δ1(I). That is a∈I or b∈I. So I is primary.
Conversely letI be a primary hyperideal. Assume that ab∈I. Since I primary hyperideal a∈I or b∈I, thus a∈I or b∈I=δ1(I).Remark 14.
(1) Ifδ and γ are two hyperideal expansions and δ(I)⊆γ(I) for each hyperideal I, then every δ-primary hyperideal is also γ-primary. Thus, in particular, a prime hyperideal is δ-primary for every δ hyperideal expansion. Let I be δ-primary. Assume that, for all a,b∈R, ab∈I and a∉I. Since I is δ-primary and δ(I)⊆γ(I) and b∈δ(I)⊆γ(I), thus b∈γ(I).
(2) Given two hyperideal expansionsδ1 and δ2, define δ(I)=δ1(I)∩δ2(I). Then δ is also a hyperideal expansion. Since δ1 and δ2 are hyperideal expansions I⊆δ1(I) and I⊆δ2(I), then I⊆δ1∩δ2(I)=δ(I) and I⊆δ(I).
LetP and Q be any hyperideals of R and P⊆Q. Thus δ1(P)⊆δ1(Q) and δ2(P)⊆δ2(Q). Finally we find δ1(P)∩δ2(P)⊆δ1(Q)∩δ2(Q).
(3) Letδ be a hyperideal expansion. Define Eδ(P)=⋂{J∈Id(R)∣P⊆J,Jisδ-primary}. Then Eδ is still a hyperideal expansion.
For allP∈Id(R), we show that P⊆Eδ(P) for any K,L∈Id(R), if K⊆L; then Eδ(K)⊆Eδ(L). By the definition of Eδ(P), we conclude that P⊆Eδ(P). For any K,L∈Id(R), if K⊆L, then the δ-primary hyperideals which contain L contain also K. In addition, there may be δ-primary hyperideals which contained L but did not contain K. Hence, we conclude that Eδ(K)⊆Eδ(L).Lemma 15.
A hyperidealP is δ-primary if and only if for any two hyperideals I and J, if IJ⊆P and I⊈P then J⊆δ(P).Proof.
LetP be δ-primary. Suppose IJ⊆P and I⊈P, but J⊈δ(P), and then we can choose a∈I-P and b∈J-δ(P). Then ab∈IJ⊆P but a∉P and b∉δ(P). This contradicts the assumption that P is δ-primary.
Conversely, if the condition is satisfied, for any two elementsa and b, suppose ab∈P and a∉P. Then (a)(b)⊆P and (a)⊈P. So (b)⊆δ(P). Hence b∈(b)⊆δ(P) implies b∈δ(P). Thus P is δ-primary.Recall that ifI and J are ideals of a commutative ring R, then their ideal quotient denotes (I:J) defined by (I:J)={r∈R∣rJ⊂I}. We recall also ideal quotient (I:J) is itself an ideal in R.Theorem 16.
Letδ be a hyperideal expansion. Then (1)
ifP is a δ-primary hyperideal and I is a hyperideal with I⊈δ(P), then (P:I)=P,(2)
for anyδ-primary hyperideal P and any subset N of the R, (P:N) is also δ-primary.Proof.
(1) From the definition of(P:I), for all x∈I·(P:I), x∈∑i=1n‍aipi, ai∈I and pi∈(P:I). Since P is a hyperideal x∈∑i=1n‍aipi⊆P, then we get x∈P. In other words I·(P:I)⊆P. Since P is δ-primary, if I⊈δ(P) then (P:I)⊆P.
Conversely, sinceP⊆(P:I) then (P:I)=P.
(2) For alla,b∈R, assume that ab∈(P:N) and a∉(P:N). Then there exists a n∈N such that an∉P. But anb=abn∈P. Thus b∈δ(P). Since δ(P)⊆δ(P:N), b∈δ(P:N). By this way we get that (P:N) is δ-primary.Theorem 17.
Ifδ is a hyperideal expansion such that δ(I)⊆δ1(I) for every hyperideal I, then, for any δ-primary hyperideal P, δ(P)=δ1(P).Proof.
For allI hyperideals, since δ(I)⊆δ1(I), δ(P)⊆δ1(P).
Conversely, leta∈δ1(P). We show that a∈δ(P).
Then there existsk which is the least positive integer k with ak∈P. If k=1 then a∈P⊆δ(P). If k>1 then ak-1a∈P. But ak-1∉P, so a∈δ(P). Hence δ1(P)⊆δ(P) and δ(P)=δ1(P).


## 4. Expansions with Extra Properties

In this section we investigateδ-primary hyperideals where δ satisfy additional conditions and prove more results with respect to such expansions.Definition 18.
A hyperideal expansionδ is intersection preserving if it satisfies (2)δI∩J=δI∩δJfor  anyI,J∈IdR.An expansion is said to be global if for any hyperring homomorphism f:R→S(3)δf-1I=f-1δI∀I∈IdS.The expansions δ0 and δ1 are both intersection preserving and global.For anyI,J∈Id(R), δ0(I∩J)=I∩J=δ0(I)∩δ0(J)=I∩J:(4)δ1I∩J=I∩J=δ1I∩δ1J=I∩J.Andδ0(f-1(I))=δ(I)=I=f-1(I).δ
1
(
f
-
1
(
I
)
)
=
δ
1
(
I
)
=
I
=
f
-
1
(
I
). Thus δ0 and δ1 are both intersection preserving and global.Theorem 19.
Letδ be an intersection preserving hyperideal expansion. If Q1,Q2,…,Qn are δ-primary hyperideals of R and P=δ(Qi) for all i, then Q=⋂i=1nQi is δ-primary.Proof.
Letδ be an intersection preserving hyperideal expansion. If x,y∈Q and x∉Q, then x∉Qk for some k. But xy∈Q⊆Qk and Qk is δ-primary, so y∈δ(Qi). But δ(Q)=δ(⋂i=1nQi)=⋂i=1nδ(Qi)=P=δ(Qk). Thus y∈δ(Q). So Q is δ-primary.Definition 20.
LetR be a hyperring and δ be a hyperideal expansion. If for an a∈R, a∈δ{0} then a is called δ nilpotent.Note thatδ0 nilpotent element of a ring is the zero element of the ring. Also δ1 nilpotent elements are exactly the ordinary nilpotent elements.Theorem 21.
Letδ be a global expansion. Let a hyperideal I of R be δ-primary and then every zero divisor of the quotient hyperring R/I is δ nilpotent.Proof.
LetI be δ-primary. If r~=r+I is a zero divisor of R/I, then there is a s~=s+I≠I with r~s~=rs+I=I. This means that rs∈I and s∉I. Since I is δ-primary so r∈δ(I); that is, r~∈δ(I)/I.
Letq:R→R/I be the natural quotient hyperring homomorphism. As δ is global, we have δ(I)=δ(q-1(0R/I))=q-1(δ(0R/I)).
Sinceq is onto, so δ(I)/I=q(δ)(I)=δ(0R/I). Hence we get r~∈δ(0R/I), so r~ is δ nilpotent.Theorem 22.
Ifδ is global and f:R→S is a hyperring homomorphism, then, for any δ-primary hyperideal I of S, f-1 is a δ-primary hyperideal of R.Proof.
Leta,b∈R with ab∈f-1(I). If a∉f-1(I) then f(a)f(b)∈I but f(a)∉I. So, as I is δ-primary, f(b)∈δ(I). So b∈f-1(δ(I))=δ(f-1(I)). Hence f-1(I) is δ-primary.Theorem 23.
Letf:R→S be a surjective hyperring homomorphism. Then a hyperideal I of R that contains ker(f) is δ-primary hyperideal of S.Proof.
Iff(I) is δ-primary, then by I=f-1(f(I)) and Theorem 22, I is δ-primary. Now suppose f(I) is δ-primary. If a,b∈S and ab∈f(I) and a∉f(I), then there are x,y∈R with f(x)=a,f(y)=b. Then f(xy)=f(x)f(y)=ab∈f(I) implies xy∈f-1(f(I))=I and f(x)=a∉f(I) implies xy∈f-1(f(I))=I and f(x)=a∉f(I) implies x∉I.
Soy∈δ(I), and hence b=f(y)∈f(δ(I)). Now one only needs to prove f(δ(I))=δ(f(I)). But this follows directly from δ(I)=δ(f-1(f(I)))=f-1(δ(f(I))) and that f is surjective.The following theorem does not need a proof because it is a consequence of Theorems22 and 23.Correspondence Theorem forδ-Primary Hyperideals.Letfbe a hyperring homomorphism of a hyperringRonto a hyperringSand letδbe global hyperring expansion. Thenfinduces a one-one inclusion preserving correspondence betweenδ-primary hyperideals ofRcontainingker⁡fand theδ-primary hyperideals ofSin such a way that ifIis aδ-primary hyperideal ofRthatIcontainsker⁡f, thenf(I)is the correspondingδ-primary hyperideal ofS, and ifJis aδ-primary hyperideal ofS, thenf-1(J)is the correspondingδ-primary hyperideal ofR.



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