Algebra
Volume 2014 (2014), Article ID 392902, 8 pages
http://dx.doi.org/10.1155/2014/392902

Research Article

Some Properties of Multiplicative -Rings of Polynomials over Multiplicative Hyperrings

Utpal Dasgupta

Department of Mathematics, Sree Chaitanya College, Habra-Prafullanagar, 24 Parganas (North), West Bengal 743268, India

Received 22 May 2014; Accepted 30 September 2014; Published 27 October 2014

Academic Editor: Andrei V. Kelarev

Copyright © 2014 Utpal Dasgupta. This 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.

Abstract.

The set of all polynomials , over a multiplicative hyperring , form a commutative group with respect to the component-wise addition (+) of the polynomials. For polynomials in , is a set of polynomials whose th components are chosen from the set , where and are the th and the th components of and , respectively. A multiplicative hyperring is polynomially structured if the hyperstructure is a multiplicative -ring. The purpose of the paper is to study the properties of the multiplicative -ring , corresponding to those of a polynomially structured multiplicative hyperring .

1. Introduction

-structures [1] are introduced by Vougiouklis, at the Fourth AHA congress in the year of 1990. Since then, the study of -structure theory has been approached in several directions by many researchers (see [2–5]). The essence of the notion of -structures is to generalize the well-known algebraic hyperstructures (such as hypergroup, hyperring, and hypermodule), simply by replacing some or all axioms of the respective hyperstructures by the corresponding weak axioms. The -structure of our initial concern is multiplicative -ring, studied in [6, 7], which is a commutative group along with a hyperoperation such that (i) is an -semigroup [3, 8] (i.e., a hyperstructure in which is weak associative in the sense that , for all ) and (ii) is weak distributive with respect to + (i.e., and , for all . A multiplicative -ring is commutative if , for all . The identity element of the group is said to be absorbing in the multiplicative -ring if , for all . A nonempty finite subset of a multiplicative -ring is called an identity set (or -set, in short) [9] of if (i) for at least one and (ii) for any , . An element of is called a hyperidentity of if the set is an -set of .

Unlike a ring, the equality of the set-expressions , , and does not hold in general on a multiplicative -ring for any . In fact, if is the ring of integers and if is a hyperoperation on , defined by , for all , then is a commutative multiplicative -ring, in which , , and .

A multiplicative -ring is said to satisfy the condition () if the set equality (called the condition () [9]) holds true for any two elements and of . Let be a ring and a hyperoperation on , defined by , for all . Then, is a multiplicative -ring with condition ().

We consider now an -structure in which (i) is a commutative group, (ii) is an -semigroup, and (iii) is semidistributive across the operation + (i.e., and , for all ). This -structure is clearly a multiplicative -ring and we thus call it a semidistributive multiplicative -ring. Henceforth, throughout the paper, a multiplicative -ring wherever considered will always be assumed to be a semidistributive multiplicative -ring with the condition ().

In 1982, the notion of multiplicative hyperring is inducted in the study on hyperring theory by Rota, which is subsequently investigated in [10–14]. A commutative group endowed with an associative hyperoperation is called a multiplicative hyperring [15] if (i) is semidistributive across the operation + on and (ii) satisfies the condition () for elements in . An associative hyperoperation is eventually weakly associative and thus a multiplicative hyperring is, eventually, a semidistributive multiplicative -ring with condition ().

Procesi Ciampi and Rota define in [16] polynomials over multiplicative hyperring as follows: let be a multiplicative hyperring with absorbing zero and let be any symbol out of . Then, a polynomial in is an expression of the form , (, ), in which + is a connective and only a finite number of the ’s (called the coefficients of in ) are different from zero of . The degree of a polynomial (in short ) is a nonnegative integer such that and , for all. A polynomial over a multiplicative hyperring will be written as when and only when . For an integer and any , ,  , we write the polynomials and simply as and , respectively. Denote by the set of all polynomials in over and define on a binary operation + and a hyperoperation as follows: for any two polynomials and , from , and , where, for any , the juxtaposition means the set . The purpose of the present paper is to study the properties of the hyperstructure , in connection to those of a particular class of multiplicative hyperrings, called polynomially structured multiplicative hyperrings which we describe formally in the following section.

2. Polynomially Structured Multiplicative Hyperring

It is asserted in [16] that for a multiplicative hyperring , the hyperstructure is always a multiplicative hyperring. But we note here that, given a multiplicative hyperring with absorbing zero, the hyperoperation (as is defined in Section 1) does not necessarily induce a multiplicative hyperring structure over the group of polynomials . In fact, we have the following example.

Example 1. Let be the ring of integers and the multiplicative hyperring where, for any ,   (denoting the product of elements in the ring simply by the juxtaposition ). Consider three polynomials , , and over the multiplicative hyperring .
Then, the set of coefficients of in the polynomials belonging to is . Again, the set of coefficients of in the polynomials belonging to is . By a tedious but routine calculation, one can see that , whereas . So, there are some polynomials in , the coefficient of in each of which is an element of the set . These polynomials do not belong to   (since ). Thus, . So, there is no question of claiming to be a multiplicative hyperring.

However, we observe that it is possible to construct a multiplicative hyperring corresponding to which the hyperstructure turns out to be a multiplicative -ring, if not a multiplicative hyperring at all.

Let us consider a commutative group . Suppose that is such that, for any and , , where denotes the image of under and is simply the mapping composition. Define a hyperoperation on by stating that . Then we have the following.

Lemma 2. is a multiplicative hyperring with absorbing zero.

Proof. Let . Then,   for some and   for some ⇒ (since ). So, . The reverse inclusion can also be shown to be true by adopting similar arguments. Hence, . Now, (for some ) . So, . Again, (for some ) (since ) . So, . Moreover, (since ) and (since ) . Thus, is a multiplicative hyperring. Finally, if denotes the identity element of the group , then (since for , ) and also (since, for is the zero homomorphism from to ). Thus, is absorbing in the multiplicative hyperring .
The multiplicative hyperring , defined in Lemma 2, is called a multiplicative -hyperring (of course if such a set exists for the group ). That a multiplicative -hyperring exists is evident in the following example.

Example 3. Let be a ring. Then, as is shown in [9, 13], is a multiplicative hyperring with absorbing zero, where with and is the -hyperoperation [8, 17] on the semigroup ; that is, for all . Now, for each and , we define a mapping by stating that , for all . Then . Thus, corresponding to each , we have a mapping , given by , for any . Then, . Moreover, for any and , ; that is, . Thus, the hyperoperation   for is defined on the group . Note that, for any , . Thus, is a multiplicative -hyperring, for .

Proposition 4. For a multiplicative -hyperring , the hyperoperation induces a multiplicative -ring structure on the group of polynomials over .

Proof. Let , , and be three polynomials in . Then, and . Now, we choose and fix an element . Then, for each ,

Again,   (since α is a homomorphism) = (since ), (since is a homomorphism) = . Clearly then, for each ,

Hence, . It is shown in [16] that, for any multiplicative hyperring the hyperoperation defined on is semidistributive over the operation + on and also satisfies the condition for any two polynomials in . Thus, for the multiplicative -hyperring , the hyperstructure is a multiplicative -ring (with absorbing zero ).
We call a multiplicative hyperring with absorbing zero polynomially structured if is a multiplicative -ring. The class of multiplicative -hyperrings is a subclass of the class of polynomially structured multiplicative hyperrings (by Proposition 4). Throughout the rest of the paper, will stand for a polynomially structured multiplicative hyperring.

3. Polynomials over Integral Hyperrings

An element of a multiplicative -ring is a left (resp., right) divisor of zero in if there exists (resp., ) such that (resp., ) and a divisor of zero in if it is either a left or a right divisor of zero in . An element of is a left (resp., right) strong divisor of zero in if there exists (resp., ) such that (resp., ) and a strong divisor of zero in if it is either a left or a right strong divisor of zero in .

Definition 5. A multiplicative -ring is called an integral -ring if there is no strong divisor of zero in it. A commutative integral -ring is an -domain. A strong integral -ring is a multiplicative -ring in which there is no divisor of zero. A strong -domain is a commutative strong integral -ring. We call an integral -ring (resp., an -domain)   simply an integral hyperring (resp., a hyperdomain) [13], when the -ring is a multiplicative hyperring.

Before entering into the study of the multiplicative -ring of polynomials over integral hyperring and hyperdomain, let us go through the following useful observations.

Remark 6. (a) For any polynomially structured multiplicative hyperring , the identity element of the group is absorbing in the multiplicative -ring .
(b) is commutative if is a commutative multiplicative hyperring.
(c) The mapping , defined by, for all , and , is a strong monomorphism. In fact, for any , . Thus can be identified with its isomorphic image in and for any , we can write the polynomial simply as and the zero polynomial as .
(d) If is an -set of , then the set is an -set in the multiplicative -ring , where, for any , denotes the set . In fact, for any , we have that = . Then, since , we have that , identifying with . Similarly, one can see that .
On the other hand, if, for some , is an -set in , then is an -set in . In fact, for any , = . Similarly, from , one may arrive at .

This is clear from Remark 6 (d) that, for any hyperidentity of , the polynomial is a hyperidentity in the multiplicative -ring . Is every hyperidentity of of the form for some hyperidentity of ? Following is an example of a multiplicative hyperring such that has a hyperidentity for any hyperidentity of .

Example 7. Let be the ring of integers and . Then is a commutative polynomially structured multiplicative hyperring (as is shown in Example 3). Denote the multiplicative hyperring by . Consider a polynomial . Then, for any , we see that and (for . Clearly, and for any ,   since, . Thus, and so is a hyperidentity in , which is not in the form for any hyperidentity of .

Remark 8. Let be a hyperidentity in the multiplicative -ring . Then, from Remark 6 (d), is a hyperidentity in the multiplicative hyperring .

Proposition 9. Let be a strong integral hyperring. Then every hyperidentity in the multiplicative -ring is of the form , for some hyperidentity of .

Proof. Suppose that is a hyperidentity in the multiplicative -ring . Then, by the Remark 8, is a hyperidentity in . Now, let be arbitrary and , where and , for all . Then, , whereby , for all , whence for all (since is a strong integral hyperring and ). Thus, , where is a hyperidentity in .

Definition 10. If , then the smallest integer such that is called the order of and is denoted by . The order of is defined to be zero. For a nonempty set , the smallest element in the set does exist and is called the order of , being denoted by ; that is, .

In the next proposition we will find some properties of , for some . For that, it is necessary, at this point, to frame some suitable notations corresponding to different types of hyperproducts of elements in the multiplicative -ring . Indeed, in any multiplicative -ring (which is not a multiplicative hyperring), the expression like () bears no connotation in , unless the parentheses “(” and “)” are meaningfully inserted in. Note that the following two expressions

are meaningful, called the finite hyperproducts of type and type and written in notations, respectively, as and .

Proposition 11. (i) For any ,

(ii) If is a strong hyperdomain, then the implication that

for all holds true for any with .

Proof. (i) Let and . Then, , where ,   for and , where ,   for . So, for any , we see that, whenever ,   (since is absorbing in and , ). Thus, if , then for any , we have that . So, .
(ii) Now suppose that is a strong hyperdomain with absorbing zero. Consider two polynomials . Let and . Then, , where , for and , where , for . So, (since is a strong hyperdomain) and also (since for and for ). Thus, for any , (since ) and , for all with . Thus, for any and so . Hence, , for all . So, the implication is true for (noting that ). Suppose that, for some integer , the implication holds true for each value of ranging from to and take any for . Then, , for all . Now, let be arbitrary. Then, for some . So, since . Thus, , for all .
Hence, by strong induction the implication follows for any with .

Corollary 12. If is a strong hyperdomain then the implication

for all holds true for any with .

Proof. Since the multiplicative hyperring is a hyperdomain, it is commutative and so is also a commutative multiplicative -ring. Hence, for any (), we have that . Hence, the assertion follows straight from Proposition 11.

Proposition 13. If the multiplicative hyperring is a strong hyperdomain, the multiplicative -ring is a strong -domain.

Proof. being a strong hyperdomain is a commutative multiplicative hyperring. Thus is a commutative multiplicative -ring. Again, since is absorbing in , is also absorbing in . Thus we take . Then, by Proposition 11, . Hence . So, is a strong -domain.

4. -Ideals in

A subgroup of the group is called a left (resp., right) -ideal of a multiplicative -ring if, for any and , (resp., ). is an -ideal of if it is both a left and a right -ideal of .

We call an -ideal of a multiplicative -ring simply a hyperideal when is a multiplicative hyperring. The notion of a typical hyperideal in a multiplicative hyperring, called -ideal, is introduced in [18] to study prime and primary hyperideals of multiplicative hyperrings. A hyperideal of a multiplicative hyperring is a -ideal if for any ,  , where , . Following is the definition of a -ideal in an arbitrary multiplicative -ring.

Definition 14. A left (resp., right)  -ideal of a multiplicative -ring is called a left (resp., right)  -ideal if for any type hyperproduct   (resp., type hyperproduct ) of elements , we have that   (resp., ). An -ideal of a multiplicative -ring is called a -ideal if it is a left as well as a right -ideal in .

We write -ideal() (resp., ) to denote the set of all (resp., proper) -ideals of a multiplicative -ring . In a commutative multiplicative -ring , every left -ideal is a right -ideal and vice versa, since commutativity implies the equality , for any . Note that in a multiplicative hyperring (even if it is not commutative) a hyperideal is a left -ideal (as an -ideal) if and only if it is a right -ideal.

Proposition 15. If is a strong hyperdomain, then the set is nonempty.

Proof. For each , let . Then is a subgroup of the group (since for any ,   and , when ). Let and . If or , then (since is absorbing in ). So, let and . Then, by Proposition 11(ii), for all , . Thus, . Hence, is an -ideal of (since is a commutative multiplicative -ring). Let be a type- hyperproduct of elements of the -ring , such that . Then, for each (since is absorbing in ). Thus, , for all . Now, since , so there exists such that . Then, for any , . Hence, by definition of , and so is a left -ideal and thus a -ideal of (since is commutative). Hence, , for all .
Since the intersection of left -ideals of a multiplicative -ring is also a left -ideal of and is itself a left -ideal, so the smallest left -ideal containing a subset of , being naturally called the left -ideal generated by , exists and is, in fact, the intersection of all left -ideals containing . The left -ideal generated by a left -ideal of is called the left -closure of and is denoted by . Clearly, for a left -ideal . and , respectively, denote the right -closure of a right -ideal and the -closure of an -ideal of . For an -ideal of a commutative multiplicative -ring , . The following lemma presents a description of the set , for a left -ideal of a multiplicative -ring . The set for a right -ideal can be described dually.

Lemma 16. Let be a multiplicative -ring with an -set and let denote the set of all left -ideals of . is a mapping defined by

for any . Then for any , we have the following: (i)  , (ii)  , (iii)  , (iv)   if and only if is a left -ideal of , and (v)  left -closure of is , where for any , denotes the -times mapping composition of .

Proof. All the assertions made in this lemma can be established by adopting the arguments that are applied in proving well-known analogous results on “complete closure of a set” in semihypergroup theory (see [3, 8]).

Proposition 17. Let be a strong hyperdomain with an -set. Then, for any -ideal of the multiplicative -ring , , where for any -ideal of , .

Proof. Since is a (strong) hyperdomain, the multiplicative -ring is commutative. So, for any -ideal of , (by Lemma 16(v)). For any , , for some type- hyperproducts of elements of , satisfying . Since here is a strong -domain (by Proposition 13) with absorbing zero, we may assume that for each (since for any ). Then, for each and , . Thus, by Corollary 12, for any . Now since, for each , , so there exists such that . Then, for any , . Consequently, , for any . Now, let be arbitrary. Then for some . So, we have that . Thus, .

5. Polynomials over Multiplicative Hyperfield

A nonzero element of a multiplicative -ring with an -set is referred to be an -invertible element (or an -unit) of if, for each , , there exist , and , such that . An element of the multiplicative -ring with a hyperidentity is said to be -hyperinvertible (or an -hyperunit) in if there exist and such that .

If is an -unit (resp., -hyperunit) in a multiplicative -ring with two -sets and (resp., with two hyperidentities and ), then one can easily verify that is also an -unit (resp., an -hyperunit) in . We thus call an -unit (resp., an -hyperunit) of a multiplicative -ring simply a unit (resp., a hyperunit). Denote by and , respectively, the sets of units and hyperunits of a multiplicative -ring .

An -ideal of a multiplicative -ring is maximal in if, for any -ideal of , . For a commutative multiplicative -ring with an -set, this is immediate to observe that if and only if , for any maximal -ideal of .

Proposition 18. Let the multiplicative hyperring be commutative and contain a hyperidentity . Then, for a polynomial , if and only if .

Proof. Since is a hyperidentity in the multiplicative hyperring , is a hyperidentity in the multiplicative -ring . Now let . Then, there exist (; ) such that . So, and thus there exists such that whence (since is commutative).
Conversely, let be hyperinvertible in with respect to the hyperidentity . Then there exists such that . We assert that there is a sequence in whose th term () is inductively defined so as to satisfy the relation that

In fact, we see that there exist such that (the relation # for ). Suppose, for some , the terms are defined in such a way that each satisfies the relation (#) for . Then () is defined to be a nonempty subset of . Let . Then there exists such that (the relation # for ). Hence the assertion is true for all . Thus consider the polynomial . Then from the definition of , and for . Thus, (due to relation # and since ). So, is hyperinvertible in ; that is, .
The (left, right) -ideal of a multiplicative -ring generated by is the smallest (resp., left, right) -ideal of containing which is denoted by (resp., ) . The principal (left, right) -ideal of the multiplicative -ring generated by an element of , denoted by (resp., , ) , is the (resp., left, right) -ideal (resp., , ) of the multiplicative -ring .
If the multiplicative -ring has an -set, then for any , and .

Definition 19. A commutative multiplicative -ring with an -set is called a multiplicative -field (resp., an inversive multiplicative -field) if (resp., ). If a multiplicative -field (resp., an inversive multiplicative -field) is a multiplicative hyperring, then we call it a multiplicative hyperfield (resp., an inversive multiplicative hyperfield).

Proposition 20. Let be a polynomially structured inversive multiplicative hyperfield. Then a polynomial is hyperinvertible in if and only if  .

Proof. If is an inversive multiplicative hyperfield, then by Proposition 18, any polynomial is hyperinvertible in if and only if . Hence, the result follows.

Definition 21. A commutative multiplicative -ring is called a principal -ideal -ring if every -ideal of is a principal -ideal. A principal -ideal -ring which is a (strong)  -domain is called a principal -ideal (strong)  -domain.

Proposition 22. Let be a polynomially structured inversive multiplicative hyperfield. Then, the multiplicative -ring of polynomials over is a principal -ideal -ring.

Proof. Let be a hyperidentity of the inversive multiplicative hyperfield . Then, the polynomial is a hyperidentity in and thus (since is commutative). So, let be any proper -ideal of . If , then is the principal hyperideal (since, for having absorbing zero, is absorbing in ). Suppose that . Then, take a nonzero polynomial such that for any . Let us write . Then, and for any . Consider then the polynomial , where . Then, clearly . Also, by Proposition 18, is hyperinvertible in . Thus there exists such that . Now, . Again, (since ). So, . Hence, (since is a -ideal and every -ideal is a right -ideal). Now, , whence (since ). Consequently, (since is a left -ideal). Then, . Thus, .
Now, let be arbitrary. Suppose that . Then, and for any . By choice of from , here . So, one can define a polynomial , where for all and for all . Clearly then , whence . Thus, .

Remark 23. In a ring, an invertible element can never be a divisor of zero. This not true in general for a multiplicative hyperring. In fact, on the commutative group of integers , if we define a hyperoperation by stating that , for all , then is a commutative multiplicative hyperring with a hyperidentity . Every nonzero element of is a zero divisor and are, in particular, hyperunits of . To get a parity with the ring theory in this regard, we perceive the notion of strong hyperinvertibility of an element of a multiplicative hyperring.

Definition 24. A hyperinvertible element of a multiplicative hyperring with a hyperidentity is said to be strongly hyperinvertible (or a strong hyperunit), if it is not a zero divisor in that multiplicative hyperring. A commutative multiplicative hyperring with absorbing zero and a hyperidentity is said to be a strongly inversive multiplicative hyperfield if each of its nonzero elements is a strong hyperunit.

Example 25. Let , and . Then, with respect to usual addition + of reals, is a commutative group, with identity . On , is a hyperoperation defined by

where and .
Then, is a strongly inversive multiplicative hyperfield which is polynomially structured.

Definition 26. A local (-local) multiplicative -ring is a commutative multiplicative -ring with an -set, which has a unique maximal -ideal (resp., -ideal).

Proposition 27. Let be a polynomially structured strongly inversive multiplicative hyperfield. Then, (i)   is a principal -ideal strong -domain, (ii)  for any , there exist a hyperinvertible element and such that , where is a hyperidentity in , and (iii)   is a local as well as a -local multiplicative -ring.

Proof. (i) Here is a strongly inversive multiplicative hyperfield. So, is a strong hyperdomain. Thus, by Proposition 13, the multiplicative -ring is a strong -domain. Again, by Proposition 22, is a principal -ideal -ring (since is an inversive multiplicative hyperfield). So, by Definition 21, is a principal -ideal strong -domain.
(ii) Let and . Then, . So, for any , there exist ;   such that . Thus, for some . Since is a strong -domain (by Proposition 13) with absorbing zero, so . Hence, without any loss of generality (since for any ), we may assume that for any (since ). Then, for any and , . So, by Proposition 11(ii), and so . Thus . So, by Proposition 17, is a nonzero polynomial in the -ideal , such that for all . So, there exists an invertible element and an integer such that , where is a hyperidentity in (see proof of Proposition 22).
(iii) For any integer , since , so . Again, for any -ideal of the multiplicative -ring , if is such that for all , then , where . So, is the unique maximal -ideal in . Thus, is a local multiplicative -ring. Now, (as defined in Proposition 15). So, for any . Thus, being a -ideal of , we have (see proof of Proposition 22). Hence, is a -local multiplicative -ring.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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