Original Scientific Paper
Hyperideals of (Finite) General Hyperrings
Reza Ameri
⋆, Mohammad Hamidi and Hoda Mohammadi
Abstract
A general hyperring is an algebraic hypercompositional system(R,+,·) with two hyperoperations”+" and ”·”, such that for allx, y ∈ R, x+y and x·y are non-empty subsets of R, andR satisfies the axioms similar to a ring. We introduce and study hyperideals of a general hyperring. In this regards, we construct a connection between classical rings and general hyperrings, specifically, we extend a ring to a general hyperring in nontrivial way. Moreover, a way to construct a general hyperring from set are given.
Also, we concentrate on an important class of general hyperrings, which is called Krasner hyperrings, and discuss on their hyperideals. Finally, the set of all hyperideals of a finite general (resp. Krasner) hyperring are considered and its hyperideals are investigated.
Keywords: general hyperring, hyperideals, Krasner hyperring 2010 Mathematics Subject Classification: 20N20.
How to cite this article
R. Ameri, M. Hamidi and H. Mohammadi, Hyperideals of (finite) general hyperrings,Math. Interdisc. Res. 6(2021)257−273.
1. Introduction
The hyperstructure theory, as an extension of classical structures, introduced by F. Marty in 1934 [10]. In a hyperalgebraic system (or a hyperstructure) a hy- perproduct (or a hyperoperation) of elements is a (nonvoid) set, and so any al- gebraic system is an special case of a hyperalgebraic system. F. Marty extended
⋆Corresponding author (E-mail: rameri@ut.ac.ir) Academic Editor: Mohammad Ali Iranmanesh Received 19 January 2020, Accepted 28 December 2021 DOI:10.22052/mir.2021.240436.1269
⃝c2021 University of Kashan This work is licensed under the Creative Commons Attribution 4.0 International License.
the concept of a group to a hypergroup, and other researchers presented the hy- peralgebraic concepts such as (Krasner) hyperrings, hypermodules, hyperfields, polygroups, multirings and etc. (for more details see [7], [8]). Nowadays hyper- structures are applied in several branches of sciences, such as artificial intelligence, chemistry and (hyper) complex network [8]. Hyperring is a hypercompositional structure as generalizing of a ring, that the sum and the product of two elements is not an element but a subset. The concept of Krasner hyperring was introduced by Krasner [9], who used it as a tool on the approximation of valued fields or the second type of a hyperring as multiplicative hyperring (the multiplication is a hyperoperation, while the addition is an operation was introduced by R. Rota in 1982 [12]. Further materials regarding hyperrings and mutirings are available in the literature too [1, 2, 3, 6]. Hyperideals are one of the important tools in hyperrings and some researchers have investigated in these concepts, see [4, 11].
This paper studies the general hyperrings as a generalization of Krasner hy- perrings as well as classical rings. In this regards we give a manner to extend a finite ring to a (nontrivial) general hyperring. Indeed, we show how one can ex- tend the class of (finite) rings to a class of general hyperrings. We will proceed to construct a general hyperring from an ideal nonempty set. Finally, we introduce some class of hyperideals of a special general hyperrings as well as the class of Krasner hyperrings and investigate their properties.
2. Preliminaries
In this section, we review some concepts from hyperstructures, which we need to development our paper [1, 5]. Let n ∈ N∗, H ̸=∅ and P∗(H) = {G | ∅ ̸=G ⊆ H}. A map ϱ: Hn → P∗(H) is an n-ary hyperoperation with arityn, (n= 0, nullary hyperoperation) and(H,{ϱi}i∈I)is ahyperalgebra(|I|= 1,hypergroupoid) of type φ : I → N∗. For two hyperalgebras (H,{ϱi}i∈I) and (H′,{ϱ′i}i∈I′), if I = I′, say similar hyperalgebras. Let H be a hyperalgebra, a ∅ ̸= S ⊆ H is said to be asubhyperalgebra ofH if for any(a1, . . . , ani)∈Sni, ϱi(a1, . . . , ani)⊆ S. For similar hyperalgebras (H,{ϱi}i∈I),(H′,{ϱ′i}i∈I), a map h : H → H′ is called a homomorphism, if for any i ∈ I and any (a1, . . . , ani) ∈ Hni, we have h(ϱi((a1, . . . , ani)) ⊆ϱ′i(h(a1), . . . , h(ani)) and agood homomorphism if for any i ∈ I, and any (a1, . . . , ani) ∈ Hni, h(ϱi((a1, . . . , ani)) = ϱ′i(h(a1), . . . , h(ani)).
A hypergroupoid(H, ϱ), whereϱis an associative is said to be a semihypergroup, and a semihypergroup(H, ϱ) is called ahypergroup, if for any x∈ H, ϱ(x, H) = ϱ(H, x) =H. A hypergroup(H, ϱ)is acanonical hypergroup, if always
(i) ϱ(x, y) =ϱ(y, x),
(ii) there is a unique elemente∈H,that for anyx∈H, ϱ(e, x) =ϱ(x, e) ={x} (neutral element),
(iii) x ∈ ϱ(y, z) concludes that y ∈ ϱ(x, ϑ(z)) and z ∈ ϱ(ϑ(y), x), where ϑ is a unitary operation onH(
for anyx∈H, there is a unique elementϑ(x)∈H
i.e e ∈(ϱ(x, ϑ(x))∩(ϱ(ϑ(x), x)), ϑ(e) =e, ϑ(ϑ(x)) =x)
and is denoted by (H, ϱ, e, ϑ)or(H,+,0,−).
AKrasner hyperringis an algebraic hypercompositional(R,+,·), where (i) (R,+) is a canonical hypergroup,
(ii) (R,·)is a semigroup,
(iii) for anyr, s, t∈R, r·(s+t) =r·s+r·t and (s+t)·r=s·r+t·r, (iv) for anyr ∈R:r·0 = 0·r = 0, i.e. there is a unique element0 ∈R is an absorbing element.
Ageneral hyperring is an algebraic hypercompositional(R,+,·), where (i) (R,+) is a hypergroup,
(ii) (R,·)is a semihypergroup and
(iii) for anyx, y, z∈R: x·(y+z) =x·y+x·zand(x+y)·z=x·z+y·z.A∅ ̸=I⊆R is said to be a(right)left hyperideal, if(1)(I,+)is a subhypergroup of(R,+) and(2)(R·I⊆I)(R·I)⊆I and a hyperideal ifR·I⊆IandR·I⊆I.
3. Construction of a General Hyperring
We introduce two notions of(+)-commutative general hyperring as(·)-commutative general hyperring and generalize these notions to commutative general hyperrings.
In this regards, a general hyperring is constructed on an ideal non-empty set and is determined how to extend a ring to a general hyperring.
Definition 3.1. Assume (R,+,·)is a general hyperring. ThenR is:
(i) additive commutative or ((+)-commutative)general hyperring, if it is com- mutative with respect to the additive hyperoperations”+",
(ii) multiplicative commutative or ((·)-commutative) general hyperring, if it is commutative with respect to the multiplicative hyperoperation ”·”, (iii) commutative general hyperring, if it is commutative with respect to the both
hyperoperations” + ”and”·”.
Now, we introduced a manner to construct a general hyperring from a given ring by using a normal subgroup of its multiplicative semigroup.
Theorem 3.2. Suppose(R,+, .)is a ring andN be a subgroup of its multiplicative semigroup. The there are hyperoperations +′ and ·′ on the quotient space R/N, that the hyperstructure(R/N,+′,·′)is a general hyperring.
Proof. For all xN, yN ∈ R/N define “+′” and “·′” on R/N by xN +′ yN = {xN, yN}andxN.′yN ={(x.y)N, xN, yN}.Now it is easy to see that(R/N,+′,·′) is a general hyperring.
Theorem 3.3. Assume (R,+, .) is a ring and N is a subgroup of its multi- plicative semigroup. Then there are hyperoperations “+′” and “·′” on R/N, that (R/N,+′,·′)is a general hyperring.
Proof. For allxN, yN ∈R/N define“+′” and“·′” onR by
xN+′yN={0N} ∪ ∪
z∈(x+y)N
zN andxN.′yN=
{0N} ∪ ∪
z∈(x.y)N
zN x̸=y
∪
z∈(x.y)N
zN x=y.
Letx∈R. ThenxN+′R/N={0N}∪∪
y∈R
(xN+′yN) ={0N}∪∪
y∈R
∪
z∈(x+y)N
zN = R/N. Since(R,+, .)is a ring, we get that(R/N,+′)is a hypergroup and(R/N,·′) is a semihypergroup. Thus distributive property hold, and hence,(R/N,+′,·′)is a general hyperring.
Now, a way to construct a commutative general hyperring on ideal non-empty set are given.
Theorem 3.4. Suppose R is a non empty set. Then there are hyperoperations
“+” and“·” on R, that (R,+,·)is a commutative general hyperring.
Proof. For|R|= 1, it is done. Let|R| ≥2. Then for anyx, y∈R define“+” and
“·” onR by
x+y=x·y={x, y}.
Obviously,(R,+,·)is a nontrivial commutative general hyperring.
At the next theorem, a way to construct a (nontrivial)general hyperring on an ideal ring is given.
Theorem 3.5. Assume (R,+,0,·) is a commutative ring. Then there are hyper- operations“+′” and“·′”on R, that(R,+′,0,·′)is a general hyperring.
Proof. If|R|= 1, it is clear. Let |R| ≥2. Then for anyx, y∈R,define “+′” and
“·′” onRby
x+′y={x, y, x+y} andx·′y={x·y,0}. One can verify that(R,+′,·′)is a general hyperring.
Example 3.6. (i) LetR={Z(a, b)|a, b∈Z2}, whereZ(a, b) =a+bi. Then by Theorem 3.5,(R,+′,·′)is a commutative general hyperring of cardinal8 as follows:
Z(a, b) +′Z(c, d) ={Z(a, b), Z(c, d), Z(a+c, b+d)|a, b, c, d∈Z2}}, Z(a, b)·′Z(c, d) ={Z(ac−bd, ad+bc),0| a, b, c, d∈Z2}}.
(ii) LetR={ [a b 0 c ]
|a, b, c∈Z2
}.
Then in virtue Theorem 3.5, the hyperstructure(R,+′,·′)is a non-commutative general hyperring of cardinal16as follows:
[a b 0 c ]
+′ [d e
0 f ]
={ [a b
0 c ]
, [d e
0 f ]
,
[a+d b+e 0 c+f ]
|a, b, c, d, e, f ∈Z2}}, [a b
0 c ]
·′ [d e
0 f ]
={
[ad ae+bf
0 cf
] ,
[0 0 0 0 ]
|a, b, c, d, e, f ∈Z2}}.
Now, a way to construct a class of general hyperrings on the cyclic groupZn, for everyn∈Nare given.
Theorem 3.7. Let n ∈ N. Then there is a hyperoperation "⊕" on Zn, that (Zn,⊕)is a hypergroup.
Proof. Fixa∈Zn, wherea̸= 0. Then for anyx, y∈Zn, define"⊕"as follows:
x⊕y=x+ay={x+y, x+y+a}. Let x∈ Zn. Then x⊕Zn = ∪
y∈Zn
(x⊕y) = ∪
y∈Zn
a∈Zn
{x+y, x+y+a} = Zn. Since
(Zn,+) is an abelian group, it get that(Zn,⊕)is a hypergroup.
By Theorem 3.5, a general way to constructed a general hyperring from an ideal ring was introduced. In the following, a different way from that in Theorem 3.5, to construct a general hyperring for the finite ring (Zn,+,·), whence n∈ N are presented.
Theorem 3.8. Assume n is an even. Then there are hyperoperations “⊕” and
“⊙”, that(Zn,⊕,⊙)is a general hyperring.
Proof. Fix0̸=a∈Zn, where2a= 0. By Theorem 3.7,(Zn,⊕)is a hypergroup.
Then for anyx, y∈Zn. Define⊙as follows:
x⊙y=x·ay={xy, xy+a}. Letx, y, z∈Zn. Then
(Associative):
x⊙(y⊙z) =x⊙ {yz, yz+a}={xyz, xyz+a, x(yz+a), x(yz+a) +a}and
(x⊙y)⊙z={xy, xy+a} ⊙z={xyz, xyz+a,(xy+a)z,(xy+a)z+a}. Now,x⊙(y⊙z) = (x⊙y)⊙z, impliesxa=za orxa=za+aandza=xa+a.
Thus(Zn,⊙)is a semihypergroup if and only if2a= 0.
(Distributive):
x⊙(y⊕z) = x⊙ {y+z, y+z+a}
= {x(y+z), x(y+z) +a, x(y+z+a), x(y+z+a) +a}, and
x⊙y⊕x⊙z = {xy, xy+a}+{xz, xz+a}
= {xy+xz, xy+xz+a, xy+xz+ 2a, xy+xz+ 3a}. Now, 2a= 0 implies thatka= 0 andk′a=a, where kis an even andk′ is an odd. Thus
x⊙(y⊕z) =x⊙y⊕x⊙z.
Therefore,(Zn,⊕,⊙)is a general hyperring.
Remark 1. In Theorem 3.8, ifnis odd, then(Zn,⊙)is not a semihypergroup, and so(Zn,⊕,⊙)is not a general hyperring.
Example 3.9. By Theorem 3.8,(Z6,⊕,⊙)is a general hyperring by the following hyperoperations:
+3 0 1 2 3 4 5
0 {0,3} {1,4} {2,5} {3,0} {4,1} {5,2} 1 {1,4} {2,5} {3,0} {4,1} {5,2} {0,3} 2 {2,5} {3,0} {4,1} {5,2} {0,3} {1,4} 3 {3,0} {4,1} {5,2} {0,3} {1,4} {2,5} 4 {4,1} {5,2} {0,3} {1,4} {2,5} {3,0} 5 {5,2} {0,3} {1,4} {2,5} {3,0} {1,4}
,
·3 0 1 2 3 4 5
0 {0,3} {0,3} {0,3} {0,3} {0,3} {0,3} 1 {0,3} {1,4} {2,5} {3,0} {4,1} {5,2} 2 {0,3} {2,5} {4,1} {0,3} {2,5} {4,1} 3 {3,0} {0,3} {3,0} {0,3} {3,0} {0,3} 4 {0,3} {4,1} {2,5} {0,3} {4,1} {2,5} 5 {0,3} {5,2} {4,1} {3,0} {2,5} {1,4}
.
Letm∈NandR= (Zn,⊕,⊙)be the general hyperring in Theorem 3.8. Define x∈R,1x={x}, mx =x|⊕x⊕{z. . . x⊕x}
m−times
, x1 ={x}, xm =x|⊙x⊙{z. . . x⊙x}
m−times
and (−m)x=m(n−x)and(x)−m= (n−x)m.
So, the following theorem follows.
Theorem 3.10. Suppose p = 2 and 2 ≤ n. Then in the general hyperring (Zpn,⊕,⊙):
(i) if 2a= 0, then for any k∈N, we have ka={0, a}, (ii) if 2a= 0, then for any k∈N, we have ak={0, a}.
Proof. (i) By induction on k∈N, ifk is odd, then ka =a. If k is even, then ka= 0.
(ii) By Theorem 3.8, a2 = {a2, a2+a}. If a2 = x, then a2 ≡ x (modpn)}. Since 2a= 0, there is t∈N, thatx=a(a+t)and soa|x(amidsk). So by item (i), x= 0or x=a. Then by induction for anyk∈N, we get that ak ={0, a}.
Theorem 3.11. Letpbe a prime. Then there are hyperoperations“+p” and“·p”, that(Zp,+p,·p)is a general hyperring.
Proof. Fix0̸=a∈Zn. Forx, y∈Zn, define“+p” and “·p” as follows:
x+{p,a}y= {
0 x=y = 0
{x+y, x+y+a} otherwise and x·{p,a}y= {
0 x= 0 ory= 0 Zp otherwise . Letx∈Zp. Then
x+{p,a}Zp={0} ∪ ∪
y∈Zp
(x+{p,a}y) ={0} ∪ ∪
y∈Zp
a∈Zp
{x+y, x+y+a}=Zp.
Since (Zp,+,·) is a commutative ring, we get that (Zp,+p)is a hypergroup. In addition, x·{p,a}Zp is equal toZp or {0}, so (Zp,·p) is a semihypergroup. The distributive property is valid, thus(Zp,+p,·p)is a general hyperring.
4. Hyperideals of General Hyperrings
Now, we present hyperideals of a general hyperring. In particular, we determine hyperideals of finite commutative general hyperrings.
Theorem 4.1. Suppose(R,+,·)is a general hyperring and ∅ ̸=I ⊆R. Then I is a hyperideal ofR if and only if the following hold:
(i) for any x∈I, x+I=I+x=I,
(ii) for any r∈R andx∈I, we have (r·x)∪(x·r)⊆I.
Proof. Immediate by definition.
Assume (R,+,·)is a general hyperring. We symbolize the set hyperideals of R byI(R). Clearly,R ∈ I(R)̸=∅ and will call Ras a non-proper hyperideal of any general hyperring.
Example 4.2. (i) Consider general hyperrings in Example 3.6 (i). Then I(R) ={I1={0}, I2={0,1, i,1 +i}, I3={0,1 +i}}. (ii) Consider general hyperringsR, in Example 3.6 (ii). Then
I(R) = {
I1={A1}, I2={A1, A2}, I3={A1, A2, A3, A7}, I4
= {A1, A2, A4, A5}, , I5=R} , where,
A1= [0 0
0 0 ]
, A2= [0 1
0 0 ]
, A3= [0 0
0 1 ]
, A4= [1 0
0 0 ]
, A5= [1 1
0 0 ]
, A6=
[1 0 0 1 ]
, A7= [0 1
0 1 ]
and A8= [1 1
0 1 ]
. Easily,I3∪I4={A1, A2, A3, A4, A5, A7} ̸∈ I(R).
Theorem 4.3. Assume (R,+,·) is a commutative general hyperring and I, I′ ∈ I(R). Then
(i) I+I′∈ I(R).
(ii) if I∩I′̸=∅, then I∩I′∈ I(R).
Proof. (i) Clearly, I+I′ ̸= ∅. Let a ∈ I and a′ ∈ I′. Then for any z ∈ ((a+a′) +I+I′), there is b ∈ I, b′ ∈ I′, that z ∈ (a+a′) + (b+b′) = (a+b) + (a′+b′)⊆I+I′. Ifz∈I+I′be an arbitrary element inI+I′, then there are a, b, c∈I,anda′, b′, c′∈I′, thatz∈c+c′⊆(a+b) + (a′+b′) = (a+a′) + (b+b′)⊆(a+a′) + (I+I′). Hence(a+a′) + (I+I′) =I+I′.Now, for any r∈R, a∈I and a′ ∈I′, one obtains(r·(a+a′))∪((a+a′)·r) = (r·a+r·a′)∪(a·r+a′·r) =r·a+r·a′ ⊆I+I′. HenceI+I′ ∈ I(R).
(ii) It is straightforward.
Theorem 4.4. Assume (R,+,·),(S,+,·)are general hyperrings,f :R→S be a homomorphism, andI∈ I(R)andJ ∈ I(S).
(i) If f is an epimorphism, thenf(I)∈ I(S).
(ii) f−1(J)∈ I(R).
Proof. (i) Since ∅ ̸= I, we have f(I) ̸= ∅. Let f(a) ∈ f(I). Then for every f(b)∈f(I), there isa′∈I, thatb∈a+a′, and sof(b)∈f(a+a′) =f(a) + f(a′)⊆f(a) +I. Hencef(I)⊆f(a) +f(I). Ifc∈f(a) +f(I)is an arbitrary element, then there isa′∈I, thatc∈f(a)+f(a′) =f(a+a′)⊆f(I). Hence, f(a) +f(I) =f(I). Now, for anys ∈S and f(a)∈ f(I), there is r ∈ R, that
(s·f(a))∪(f(a)·s) = (f(r)·f(a))∪(f(a)·f(r)) = (f(r·a))∪(f(a·r))
= f(r·a∪a·r)⊆f(I).
(ii) It is straightforward.
4.1 Hyperideals of R = ( Z
n, ⊕ , ⊙ )
At the following the hyperideals of a commutative general hyperringR= (Zn,⊕,⊙) are obtained and it is shown that they are depended to the divisors elements, say a, wherea∈Zn and2a= 0.
Theorem 4.5. Assume pis a prime. ThenI(Zp,+p,·p) ={{0},Zp}.
Proof. Obviously, I = {0} ∈ I((Zp,+p,·p). Let {0} ̸= I ∈ I((Zp,+p,·p) be a hyperideal and0̸=x∈I. Since for any0̸=y∈Zp,Zp=x.py⊆I, it obtain that I=Zp. Thus|I(Zp,+p,·p)|= 2.
Example 4.6. Leta= 2. Then forn= 4,R= (Zn,+n,·n)is a general hyperrings andI(Zn,+n,·n) ={{0},Zn}and otherwiseR= (Zn,+n,·n)is not a hyperring.
Example 4.6, disprove the converse of Theorem 4.5, necessarily is not true.
Assume R = (Zn,⊕,⊙) is the general hyperring in Theorem 3.8 and x∈ R.
Define
⟨x⟩= ∪
k∈N
kx.
Thus have next results.
Theorem 4.7. Suppose2≤n is an even,a∈R andx∈R. If2a= 0, then (i) ⟨x⟩ ∈ I(Zn,⊕,⊙),
(ii) ⟨0⟩=⟨a⟩,
(iii) if x̸=aandgcd(x, a) =d, we have ⟨x⟩=⟨d⟩, (iv) I∈ I(Zn,⊕,⊙)⇔there isx∈R, thatI=⟨x⟩.
Proof. (i) Letx∈R. By definition, we have
⟨x⟩= ∪
k∈N
{kx, kx+a}.
This shows that it is a hyperideal ofR. Lety ∈ ⟨x⟩andz∈y⊕ ⟨x⟩. Thus there isk, k′∈Nandw∈ ⟨x⟩, thatz∈y⊕wand hence
z∈kx⊕k′x, z∈kx⊕k′x+aandz∈kx+a⊕k′x.
Since2a= 0, there isk′′∈N, that
z∈ {k′′x, k′′x+a} ⊆ ⟨x⟩. Similarly, it implies that r∈Zn andy∈ ⟨x⟩. Thus
r⊙ ⟨x⟩ ⊆ ⟨x⟩. Therefore,
⟨x⟩ ∈ I(Zn,⊕,⊙).
(ii) By Theorem 3.10, we get that
⟨0⟩=⟨a⟩={0, a}. Therefore,I is a hyperideal as desired.
(iii) Let y ∈ ⟨x⟩. Then there is k ∈ N, that y = kx or y = kx+a. Since gcd(x, a) =dand2x̸= 0, by item(i), there is k′∈Z, thatx=k′d.
Ify=kx, then
y=kx=kk′d∈ ⟨x⟩,
and ify=kx+a, theny=kx+a=kk′d+a∈ ⟨x⟩. Hence
⟨x⟩ ⊆ ⟨d⟩.
Let y ∈ ⟨d⟩. Then there is k ∈ N, that y = kd or y = kd+a. Since gcd(x, a) =dand 2x̸= 0by(i), there isr, s∈Z, thatrx+as=d, and so rkx+aks=kd. Applying Theorem 3.10, we get thaty=krxory=krx+a.
Hence
⟨d⟩ ⊆ ⟨x⟩.
(iv) Let I ∈ I(Zn,⊕,⊙). Then I ̸= ∅ and so there is x ∈ I. Since I is a hyperideal, we get that x⊕x⊆I. By induction on k∈Nit conclude that kx⊆I, and hence
I= ∪
k∈N
{kx, kx+a}=⟨x⟩. This complete the proof.
Example 4.8. Consider the general hyperringR= (Z12,⊕,⊙). By Theorem 4.7, I(Z12,⊕,⊙) ={I1={0}, I2={0,2,4, . . . ,10}, I3={0,3,6,9}, I4={0,6}, I5=Z12}. Theorem 4.9. Let 2≤nand a∈R. If2a= 0, then
(i) |I(Zn,⊕,⊙)|=|Div(a)|+ 1, (ii) if for anyx∈R, x|a, thena∈ ⟨x⟩.
Proof. (i) Clearly {0} ∈ I. By Theorem 4.7,{0} ̸=I ∈ I if and only if there is x ∈ R that I = ⟨x⟩. Also for any x ∈ R, gcd(x, a) = d if and only if
⟨x⟩=⟨d⟩. So|I(Zn,⊕,⊙)|=|Div(a)|+ 1.
(ii) Letx∈R. By definition, we have⟨x⟩= ∪
k∈N
{kx, kx+a}. Ifx|a(xmidsa), then there isk∈N, thata=kxand soa∈ ⟨x⟩.
Corollary 4.10. Let 2 =p1, p2, . . . , pk be primes,k, α1, α2, . . . , αk ∈N andn=
∏k i=1
pαii. Then
(Zn,⊕,⊙) ={0}∪{⟨pt11pt22. . . ptjj⟩ |0≤t1≤α1−1, and for all j̸= 1, 0≤tj ≤αi}⟩. Example 4.11. Let2 = pandq, r be odd primes, m, l, k ∈Nand n=pmqlrk. Then
I(Zn,⊕,⊙) ={0} ∪ {⟨pt1qt2rt3⟩ |0≤t1≤m−1,0≤t2≤l,0≤t3≤k}⟩. Corollary 4.12. Let 2 =p1 andp2, . . . , pk be primes, k, α1, α2, . . . , αk ∈N and n=
∏k i=1
pαii. Then |I(Zn,⊕,⊙)|= 1 +α1 k+1∏
i=2
(αi+ 1).
Example 4.13. Consider the general hyperringR= (Z36,⊕,⊙). Then we have I(Z36,⊕,⊙) ={I1={0}, I2={0,2,4, . . . ,34}, I3={0,3,6, . . . ,33}, I4={0,6,12, . . . ,30}, I5={0,9,18,27}, I6={0,18} and I7=Z36}.
4.2 Hyperideals of General Hyperring R = ( Z
n, +
′, ·
′)
In this part, all hyperideals of finite commutative general hyperringR= (Zn,+′,·′) are computed and it is proved that every hyperideal of the general hyperring R= (Zn,+′,·′)is characterized by the divisors of cardinal of Zn.
Let m ∈ N and R = (Zn,+′,·′) be the general hyperring in Theorem 3.5.
Then for any x ∈ R define, 1x = {x}, mx = x|+′x+{z′. . . x+′x}
m−times
, x1 = {x}, xm =x|·′x·′{z. . . x·′x}
m−times
and⟨x⟩= ∪
k∈N
kx.With this regard to these notations and definitions, We have the following theorem.
Theorem 4.14. Letn, d∈Nandx, y∈R. Then (i) ⟨x⟩ ∈ I(Zn,+′,·′),
(ii) ⟨0⟩={0},
(iii) ⟨x⟩=⟨y⟩ ⇔gcd(x, n) = gcd(y, n) =d.
Proof. (i) Letx∈R. By definition, we have⟨x⟩= ∪
k∈N
{kx,0}and show that it is a hyperideal ofR. Lety ∈ ⟨x⟩and z ∈y+′⟨x⟩. Thus there isk, k′ ∈N and w ∈ ⟨x⟩ that z ∈ y+′w and so z ∈ kx+′k′x, z ∈ kk′x and z ∈ ⟨0⟩. There is k′′ ∈Nthat z ∈ {k′′x,0} ⊆ ⟨x⟩. In a similar way, for any r∈Zn
andy∈ ⟨x⟩,we haver+′⟨x⟩ ⊆ ⟨x⟩.Hence ⟨x⟩ ∈ I(Zn,+′,·′).
(ii) It is straightforward by definition.
(iii) Letz∈ ⟨x⟩. Then there isk∈N, thatz=kxorz= 0. Sincegcd(x, n) =d and by item (i), there is k′ ∈ Z, thatx =k′d. If z =kx, then z =kx = kk′d∈ ⟨x⟩and ifz= 0, thenz=k0 = 0∈ ⟨x⟩. Hence⟨x⟩ ⊆ ⟨d⟩. Letz∈ ⟨d⟩. Then there is k∈N, thatz=kdorz= 0. Sincegcd(x, n) =dand by item (i), there is r, s ∈ Z that rx+ns = d, and so rkx+nks = kd. Applying Theorem 3.10, we get that z = krx or z = 0. Hence ⟨d⟩ ⊆ ⟨x⟩. Also for gcd(y, n) =dthe proof is similarly, then⟨d⟩=⟨y⟩, there for⟨x⟩=⟨y⟩.
Example 4.15. Consider the general hyperringR = (Z45,+′,·′). By Theorem 4.14, we have
I(Z45,+′,·′) ={I1={0}, I2={0,3,6, . . . ,42}, I3={0,5,10, . . . ,40}, I4={0,9,18,27,36}, I5={0,15,30} and I6=Z45}.
Theorem 4.16. Letn∈N. Then (i) |I(Zn,+′,·′)|=|Div(n)|.
(ii) for any x, y∈R,⟨x⟩ ∩ ⟨y⟩=⟨lcm(x, y)⟩.
Proof. (i) By Theorem 4.14, I ∈ I if and only if there is sx ∈ R, that I =
⟨x⟩. Also for any x ∈ R, gcd(x, n) = d if and only if ⟨x⟩ = ⟨d⟩. Thus
|I(Zn,+′,·′)|=|Div(n)|.
(ii) Letx∈R. By definition, we have⟨x⟩= ∪
k∈N
{kx,0}. Clearly, there isk1, k2∈ Nthatlcm(x, y) =k1xandlcm(x, y) =k2y. Hencelcm(x, y)∈ ⟨x⟩∩⟨y⟩and so⟨lcm(x, y)⟩ ⊆ ⟨x⟩ ∩ ⟨y⟩.Conversely. leta∈ ⟨x⟩ ∩ ⟨y⟩. Thena= 0or there isk1, k2 ∈Nthat a=k1x=k2y. Thusx|aand y|aand solcm(x, y)|a.
Hence there isk∈Nthata=k×lcm(x, y)and so⟨a⟩ ⊆ ⟨x⟩ ∩ ⟨y⟩.
Corollary 4.17. Let n∈N. ThenI(Zn,+′,·′) ={⟨d⟩ | d∈Div(n)}. Example 4.18. Letp, q, rbe primes,m, l, k∈Nandn=pmqlrk. Then
I(Zn,+′,·′) ={⟨pt1qt2rt3⟩ |0≤t1≤m,0≤t2≤l,0≤t3≤k}⟩.
Corollary 4.19. Assume p1, p2, . . . , pk are primes,k, α1, α2, . . . , αk∈Nandn=
∏k i=1
pαii. Then |I(Zn,+′,·′)|=
∑k i=1
(αi+ 1).
Example 4.20. Consider the finite general hyperringR= (Z90,+′,·′). Based on corollary 4.17,
I(Z90,+′,·′) ={
I1={0}, I2={0,2,4, . . . ,88}, I3={0,3,6, . . . ,87}, I4={0,5,10, . . . ,85}, I5={0,9,18, . . . ,81}, I6={0,10,20, . . . ,80}, I7={0,6,12, . . . ,84}, I8={0,15,30, . . . ,75}, I9={0,18,36, . . . ,72}, I10={0,45}, I11={0,30,60}and I12=Z90
}.
4.3 Restriction of General Hyperring to Krasner Hyperring
We consider an especial of general hyperring as Krasner hyperring and construct this class upon previously section. Now, we construct a Krasner hyperrings based on quotient of a ring on normal subgroup of its multiplicative semigroup in different to Krasner way.
Theorem 4.21. Assume (R,+, .) is a ring and N is a normal subgroup of its multiplicative semigroup. Then there are hyperoperations “+′” and “·′” on R/N, that(R/N,+′,·′)is a Krasner hyperring.
Proof. For allxN, yN ∈R/N define“+′” and“·′” onR by
xN+′yN =
{xN} ∪ ∪
z∈(x+y)N
zN x̸=y
R/N x=y
xN y= 0.
andxN·′yN = (xy)N.
It is easy to verify that(R/N,+′,·′)is a commutative Krasner hyperring.
Theorem 4.22. Let Zξ =Z∪ {ξ}, where ξ̸∈Z. Then there are hyperoperation
“+ξ” and an operation “·ξ”, that(Zξ,+ξ,0,·ξ,1)is a Krasner hyperring.
Proof. Let x, y ∈ Z. Define hyperoperation “+ξ” and operation “·ξ” on Zξ as follows:
x+ξy=
{0, ξ} x=−y
x+y x, y∈Z, x̸=−y
y x= 0
ξ x=y=ξ
and x·ξy= {
x.y x, y∈Z ξ y=ξ. ,
where for any x, y ∈ Zξ, we have x+ξ y = y+ξx, x·ξ y = y·ξ x and 0·ξ x = 0. Some modifications and computations show that (Zξ,+ξ,0,·ξ,1) is a Krasner hyperring.
Theorem 4.23. LetZnξ =Zn∪{ξ}, whereξ̸∈Zn. Then there are hyperoperation
“+ξ” and an operation “·ξ” that(Znξ,+ξ,0,·ξ,1)is a Krasner hyperring.
Proof. Letx, y ∈ Zn. Define hyperoperation “+ξ” and operation “·ξ” on Znξ as
follows: x+ξy=
{0, ξ} x=−y
x+y x, y∈Zn, x̸=−y
y x= 0or (x=ξandy̸∈ {ξ})
ξ x=y=ξ
and
x·ξ y =
x.y x, y∈Zn
ξ (y=ξ, gcd(x, n)̸= 1)or(x=y=ξ) 0 x=ξ,(x, n) = 1
, where for any x, y ∈Znξ, we have x+ξy =y+ξx, x·ξy =y·ξ xand 0·ξx= 0. Some modifications and computations show that(Znξ,+ξ,0,·ξ,1)is a Krasner hyperring.
4.4 Hyperideals of Krasner Hyperrings
At the following we compute the hyperideals of some class of Krasner hyperrings based on way introduced at the previous section.
Theorem 4.24. LetZnξ=Zn∪ {ξ}, whereξ̸∈Zn andn∈N. Then (i) if nis a prime, then I(Znξ,+ξ,·ξ) ={I1={0, ξ}, I2=Znξ},
(ii) ifnis an odd non-prime, thenI(Znξ,+ξ,·ξ) ={I1={0, ξ}, I2={ai|gcd(ai
, n)̸= 1} ∪ξ, I3=Znξ},
(iii) if n is an even, then I(Znξ,+ξ,·ξ) = {I1 ={0, ξ}, I2 = {ai | gcd(ai, n) ̸= 1} ∪ξ, I3={0,n
2, ξ}, I4=Znξ}.
Proof. (i) ObviouslyI={0, ξ} ∈ I(Znξ,+ξ,·ξ). Let{0, ξ} ̸=I∈ I(Znξ,+ξ,·ξ) be a hyperideal and0̸=x∈I. Since for any0̸=y∈ZnandZn =x·ξy⊆I, it obtains thatI=Zn.
(ii) ObviouslyI ={0, ξ} ∈ I(Znξ,+ξ, .ξ). Let {0, ξ} ̸= I = {ai | gcd(ai, n) ̸= 1} ∈ I(Znξ,+ξ,·ξ)be a hyperideal thengcd(ai, n) =dconcludes thataiZ+ nZ=dZ. Because there arex, y∈Zandd=aix+ny, we get d∈aiZ+nZ and hencedZ⊆aiZ+nZ. Alsod|ai(dmidsai)andd|n(dmidsn), then aiZ⊆dZandnZ⊆dZimply thataiZ+nZ⊆dZand henceaiZ+nZ=dZ. (iii) It’s similar to (ii).
Example 4.25. For prime, odd non-prime and even numbers7,9,6, we have:
(i) ifn= 7, thenI(Z7ξ,+ξ,·ξ) ={I1={0, ξ}, I2=Z7ξ},
(ii) ifn= 9, thenI(Z9ξ,+ξ,·ξ) ={I1={0, ξ}, I2={0,3,6, ξ}, I3=Z9ξ}, (iii) ifn= 6, thenI(Z6ξ,+ξ,·ξ) ={I1={0, ξ}, I2={0,2,3,4, ξ}, I3={0,3, ξ}, I4
=Z6ξ}.
Theorem 4.26. LetZξ =Z∪{ξ}, whereξ̸∈Z. ThenI(Zξ,+ξ,·ξ) ={mZξ |m∈ N}.
Proof. Obviously for anym∈NandmZis a hyperideal ofZξ, because of for any mk∈mZ, n∈N, havenmk=mkn∈mZξ.
Example 4.27. Consider the Krasner hyperring R = (Z10,+ξ,·ξ) by Theorem 4.22. So we have
I(Z10,+ξ, .ξ) ={I1={0, ξ}, I2={0,2,4,6,8, ξ}, I3={0,5, ξ}, I4=Z10ξ}.
5. Conclusion
The notion of a general hyperring, as generalization of hyperring as well as classical ring are introduced. In this regards some ways to extend a classical ring to a general hyperring was given, also a way to construct a general hyperring from every set was introduced. Also, the hyperideals of some special classes of general hyperring such (finite) Krasner hyperring was computed. Moreover, under some certain condition hyperideals in finite general hyperrings was characterized. In particular the hyperideals of some finite Krasner hyperrings was obtained. Accordingly, This paper has been provided a good introduction to construct general hyperrings as well as to computing its hyperideals, especially for finite case. At the end, the authors propose to use a computer programming way to construct finite general hyperring and computing their hyperideals. for future works in this area.
Conflicts of Interest. The authors declare that there are no conflicts of interest regarding the publication of this article.
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Reza Ameri
School of Mathematics,
Statistic and Computer Sciences, University of Tehran,
Tehran, I. R. Iran e-mail: rameri@ut.ac.ir Mohammad Hamidi
Department of Mathematics, Payame Noor University, Tehran, I. R. Iran
e-mail: m.hamidi@pnu.ac.ir Hoda Mohammadi
Department of Mathematics, Payame Noor University, Tehran, I. R. Iran
e-mail: Hodamohamadi@student.pnu.ac.ir
