On the Jacobson Radical of an (𝑚,𝑛)-Semiring

Abstract

The notion of 𝑛-ary semimodules is introduced so that the Jacobson radical of an (𝑚,𝑛)-semiring is studied and some well-known results concerning the Jacobson radical of a ring (a semiring or a ternary semiring) are generalized to an (𝑚,𝑛)-semiring.

1. Introduction

The concept of semigroups [1] was generalized to that of ternary semigroups [2], that of 𝑛-ary semigroups [3–6], and even to that of (𝑛,𝑚)-semigroups [7]. Similarly, it was natural to generalize the notion of rings to that of ternary semirings, that of 𝑛-ary semirings, and even that of (𝑚,𝑛)-semirings.

Indeed, there were some research articles on semirings, (see, for example, [8–14]), specially on the radical of a semiring; see [15–18]. Semigroups over semirings were studied in [19] and semimodules over semirings were studied in [14]. The notion of semirings can be generalized to ternary semirings [20] and Γ-semirings [21], even to (𝑚,𝑛)-semirings [22–24]. The radicals of ternary semirings and of Γ-semirings were studied in [20, 21], respectively. The concept of (𝑚,𝑛)-semirings was introduced and accordingly some simple properties were discussed in [22–24], where the concept of radicals was not mentioned.

The notion of the Jacobson radicals was first introduced by Jacobson in the ring theory in 1945. Jacobson [25] defined the radical of 𝑅, which we call the Jacobson radical, to be the join of all quasi-regular right ideals and verified that the radical is a two-sided ideal and can also be defined to be the join of the left quasi-regular ideals.

The concept of the Jacobson radical of a semiring has been introduced internally by Bourne [15], where it was proved that the right and left Jacobson radicals coincide; thus one could say the Jacobson radical briefly. These and some other results were generalizations of well-known results of Jacobson [25].

In 1958, by associating a suitable ring with the semiring, Bourne and Zassenhaus defined the semiradical of the semiring [16]. In [18] it was proved that the concepts of the Jacobson radical and the semiradical coincide.

Iizuka [17] considered the Jacobson radical of a semiring from the point of view of the representation theory [15] without reducing it to the ring theory. The external notion of the radical was proved to be related to internal one; at the same time, it was shown that the radical defined in [17] coincides with the Jacobson radical and with the semiradical of the semiring.

In the present paper, we investigate (𝑛,𝑚)-semirings by means of 𝑛-ary semimodules so that we can define externally the Jacobson radical of an (𝑛,𝑚)-semiring, and then we establish the radical properties of the Jacobson radical of an (𝑛,𝑚)-semiring. Some necessary notions such as irreducible 𝑛-ary semimodules over an (𝑛,𝑚)-semiring are adequately defined. All results in this paper generalize the corresponding ones concerning the radical of a ring [25], of a semiring [15–18], or of a ternary semiring [20].

2. Preliminaries

We used following convention as followed by [4]: The sequence 𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑚 is denoted by 𝑥^𝑚_𝑖. Thus the following expression 𝑓𝑥₁,…,𝑥_𝑖,𝑦_𝑖+1,…,𝑦_𝑗,𝑧_𝑗+1,…,𝑧_𝑚 (1) is represented as 𝑓𝑥^𝑖₁,𝑦^𝑗_𝑖+1,𝑧^𝑚_𝑗+1. (2) In the case when 𝑦_𝑖+1=⋯=𝑦_𝑗=𝑦, then (2) is expressed as 𝑓𝑥^𝑖₁,𝑦^𝑗−𝑖,𝑧^𝑚_𝑗+1. (3) If 𝑥₁=⋯=𝑥_𝑖=𝑦_𝑖+1=⋯=𝑦_𝑗=𝑧_𝑗+1=⋯=𝑧_𝑚=𝑓(𝑎^𝑚₁), then (2) can be written as 𝑓(𝑓(𝑎^(𝑚)𝑚₁)).

Recall that an 𝑛-ary semigroup (𝑆,𝑓) is defined as a nonempty set 𝑆 with an 𝑛-ary associative operation 𝑓∶𝑆^𝑛=𝑆×⋯×𝑆_𝑛→𝑆; that is, 𝑓𝑥₁^𝑖−1,𝑓𝑥_𝑖^{𝑛+𝑖−1},𝑥^2𝑛−1_𝑛+𝑖=𝑓𝑥₁^𝑗−1,𝑓𝑥_𝑗^{𝑛+𝑗−1},𝑥^2𝑛−1_𝑛+𝑗 (4) for all 𝑥₁^2𝑛−1∈𝑆 and all 1≤𝑖<𝑗≤𝑛. Whence we may denote 𝑓(𝑓(𝑥^𝑛₁),𝑥^2𝑛−1_𝑛+1) by 𝑓(𝑥₁^2𝑛−1) briefly. Generally, we have the notation 𝑓(𝑥₁^{𝑡(𝑚−1)+1}) for each positive 𝑡 and all 𝑥_𝑗∈𝑆. Thus for positive integer 𝑘, 𝑓(𝑥^𝑘₁) is well defined if and only if 𝑘≡1mod(𝑛−1); see [7, Lemma 1.1]. An 𝑛-ary semigroup (𝑆,𝑓) is called cancellative if 𝑓𝑥₁^𝑖−1,𝑎,𝑥^𝑛_𝑖+1=𝑓𝑥₁^𝑖−1,𝑏,𝑥^𝑛_𝑖+1⟹𝑎=𝑏 (5) for all 𝑎,𝑏,𝑥_𝑗∈𝑆.

The next definition is a generalization of the concept of ternary semirings in [20] and similar to the notion of the (𝑚,𝑛)-semirings in [24].

Definition 1. A nonempty set 𝑅 together with an 𝑚-ary operation 𝑓, called addition, and an 𝑛-ary operation 𝑔, called multiplication, is said to be an (𝑚,𝑛)-semiring if the following conditions are satisfied. (1) (𝑅,𝑓) is an 𝑚-ary semigroup and (𝑅,𝑔) is an 𝑛-ary semigroup.(2)𝑔 is distributive with respect to operation 𝑓; that is, for every 𝑎₁^𝑖−1, 𝑏^𝑚₁, 𝑎^𝑛_𝑖+1∈𝑅, 𝑔𝑎₁^𝑖−1,𝑓𝑏^𝑚₁,𝑎^𝑛_𝑖+1=𝑓𝑔𝑎₁^𝑖−1,𝑏₁,𝑎^𝑛_𝑖+1,…,𝑔𝑎₁^𝑖−1,𝑏_𝑚,𝑎^𝑛_𝑖+1. (6)(3)(𝑅,𝑓) is commutative; that is, for every permutation 𝑝 of {1,2,…,𝑚} and all 𝑥^𝑚₁∈𝑅, 𝑓𝑥₁,𝑥₂,…,𝑥_𝑚=𝑓𝑥_𝑝(1),𝑥_𝑝(2),…,𝑥_𝑝(𝑚). (7)(4)  There is an element 0, called the zero of (𝑅,𝑓,𝑔), satisfying the following two properties: (4A)0 is an 𝑓-identity; that is, for every 𝑥∈𝑅𝑓(0^(𝑚−1),𝑥)=𝑥;(4B)0 is a 𝑔-zero; that is, for all 𝑥^𝑚₁∈𝑅, 𝑔(𝑥^𝑛₁)=0 whenever there exists 𝑖 such that 𝑥_𝑖=0.

It is clear that the zero of an (𝑚,𝑛)-semiring 𝑅 is necessarily unique.

Definition 2. An (𝑚,𝑛)-semiring (𝑅,𝑓,𝑔) is called additively cancellative if the 𝑚-ary semigroup (𝑅,𝑓) is cancellative and multiplicatively cancellative if the 𝑛-ary semigroup (𝑅,𝑔) is cancellative.

Recall that for an 𝑛-ary semigroup (𝑆,𝑓), a nonempty subset 𝑇 of 𝑆 is called a subsemigroup of (𝑆,𝑓) if 𝑓(𝑎^𝑛₁)∈𝑇 whenever all 𝑎^𝑛₁∈𝑇. For 𝑖≤𝑛, we call 𝑇 an 𝑖-ideal of 𝑆 if 𝑓(𝑎^𝑛₁)∈𝑇 whenever 𝑎_𝑖∈𝑇. 𝑇 is called an ideal of 𝑆 if and only if it is an 𝑖-ideal of 𝑆. See, for example, [7, Definition 1.6].

Definition 3. A nonempty subset 𝑇 of an (𝑚,𝑛)-semiring (𝑅,𝑓,𝑔) is called an 𝑛-ary subsemiring of (𝑅,𝑓,𝑔) if 𝑇 is a subsemigroup of (𝑅,𝑓) as well as a subsemigroup of (𝑅,𝑔) and an (i-)ideal of (𝑅,𝑓,𝑔) if 𝑇 is a subsemigroup of (𝑅,𝑓) as well as an (i-)ideal of (𝑅,𝑔) (where 𝑖≤𝑛). An 1-ideal is also called a right ideal and an 𝑛-ideal is also called a left ideal. An ideal 𝑇 of (𝑅,𝑓,𝑔) is called a 𝑘-ideal if 𝑓(𝑥,𝑦^𝑚₂)∈𝑇;𝑥∈𝑅 and 𝑦^𝑚₂∈𝑇 imply that 𝑥∈𝑇. An ideal 𝑇 of (𝑅,𝑓,𝑔) is called an ℎ-ideal if 𝑓(𝑥,𝑦^𝑚₂,𝑧^𝑚₂)=𝑓(𝑦₁,𝑧^𝑚₂);𝑥,𝑧^𝑚₂∈𝑅 and 𝑦^𝑚₁∈𝑇 imply that 𝑥∈𝑇.

Let 𝐴 be an ideal of (𝑅,𝑓,𝑔). Then the 𝑘-closure of 𝐴, denoted by 𝐴, is defined by 𝐴={𝑥∈𝑅∶𝑓(𝑥,𝑎^𝑚₂)=𝑎₁forsome𝑎^𝑚₁∈𝐴}. Similarly, the ℎ-closure of 𝐴, denoted by 𝐴, is defined by 𝐴={𝑥∈𝑅∶𝑓(𝑥,𝑎^𝑚₂,𝑦^𝑚₂)=𝑓(𝑎₁,𝑦^𝑚₂) for some 𝑎^𝑚₁∈𝐴 and some 𝑦^𝑚₂∈𝑅}. One can show that 𝐴 is a 𝑘-ideal and 𝐴 is an ℎ-ideal. Furthermore, it is shown that an ideal 𝐴 of (𝑅,𝑓,𝑔) is a 𝑘-ideal if and only if 𝐴=𝐴 and that 𝐴 is an ℎ-ideal if and only if 𝐴=𝐴.

Definition 4. An equivalence relation 𝜌 on an (𝑚,𝑛)-semiring (𝑅,𝑓,𝑔) is said to be a congruence relation or simply a congruence of (𝑅,𝑓,𝑔) if the following conditions are satisfied:(1)𝑎^𝑚₁𝜌𝑏^𝑚₁⇒𝑓(𝑎^𝑚₁)𝜌𝑓(𝑏^𝑚₁) for all 𝑎^𝑚₁,𝑏^𝑚₁∈𝑅, (2)𝑎^𝑛₁𝜌𝑏^𝑛₁⇒𝑔(𝑎^𝑛₁)𝜌𝑔(𝑏^𝑛₁) for all 𝑎^𝑛₁,𝑏^𝑛₁∈𝑅.

Let 𝐼 be a proper ideal of an (𝑚,𝑛)-semiring (𝑅,𝑓,𝑔). Then the congruence on (𝑅,𝑓,𝑔), denoted by 𝑘_𝐼, and defined by setting 𝑥𝑘_𝐼𝑦 if and only if 𝑓(𝑥,𝑎^𝑚₂)=𝑓(𝑦,𝑏^𝑚₂) for some 𝑎^𝑚₂,𝑏^𝑚₂∈𝐼, is called the Bourne congruence on (𝑅,𝑓,𝑔) defined by the ideal 𝐼. We denote the Bourne congruence class of an element 𝑥 by 𝑥/𝐼 and denote the set of all such congruence classes of (𝑅,𝑓,𝑔) by 𝑅/𝐼. If the Bourne congruence 𝑘_𝐼 is proper, that is, 0/𝐼≠𝑅, then we can define two operations, 𝑚-ary addition and 𝑛-ary multiplication on 𝑅/𝐼 by 𝑓(𝑎^𝑚₁/𝐼)=𝑓(𝑎^𝑚₁)/𝐼 and 𝑔(𝑏^𝑛₁/𝐼)=𝑔(𝑏^𝑛₁)/𝐼 for all 𝑎^𝑚₁,𝑏^𝑛₁∈𝑅. Then (𝑅/𝐼,𝑓,𝑔) is an (𝑚,𝑛)-semiring and is called the Bourne factor (𝑚,𝑛)-semiring.

Similarly, the congruence on (𝑅,𝑓,𝑔), denoted by ℎ_𝐼, and defined by setting 𝑥ℎ_𝐼𝑦 if and only if 𝑓(𝑥,𝑎^𝑚₂,𝑦^𝑚₂)=𝑓(𝑦,𝑏^𝑚₂,𝑦^𝑚₂) for some 𝑎^𝑚₂,𝑏^𝑚₂∈𝐼 and some 𝑦^𝑚₂∈𝑅, is called the Iizuka congruence on (𝑅,𝑓,𝑔) defined by the ideal 𝐼. We denote the Iizuka congruence class of an element 𝑥 by 𝑥[/]𝐼 and denote the set of all such congruence classes of (𝑅,𝑓,𝑔) by 𝑅[/]𝐼. If the Iizuka congruence ℎ_𝐼 is proper, that is, 0[/]𝐼≠𝑅, then we can define two operations, 𝑚-ary addition and 𝑛-ary multiplication on 𝑅[/]𝐼 by 𝑓(𝑎^𝑚₁[/]𝐼)=𝑓(𝑎^𝑚₁) and 𝑔(𝑏^𝑛₁[/]𝐼)=𝑔(𝑏^𝑛₁) for all 𝑎^𝑚₁,𝑏^𝑛₁∈𝑅. Then (𝑅[/]𝐼,𝑓,𝑔) is an (𝑚,𝑛)-semiring and we call it the Iizuka factor (𝑚,𝑛)-semiring.

The next definition is a generalization of [20, Definition 2.13].

Definition 5. A commutative 𝑚-ary semigroup (𝑀,𝑓₀) with an identity 0_𝑀 (operation 𝑓₀ to be called addition) is called a right 𝑛-ary semimodule over an (𝑚,𝑛)-semiring (𝑅,𝑓,𝑔) or simply an 𝑛-ary 𝑅-semimodule if there exists a mapping 𝑀×𝑅×⋯×𝑅_𝑛−1→𝑀 (images to be denoted by 𝑎𝑟₂⋯𝑟_𝑛 or briefly by 𝑎𝑟^𝑛₂ for all 𝑎∈𝑀 and 𝑟^𝑛₂∈𝑅) satisfying the following conditions: (1)𝑓₀(𝑎^𝑚₁)𝑟₂⋯𝑟_𝑛=𝑓₀(𝑎^𝑚₁𝑟₂⋯𝑟_𝑛) for all 𝑎^𝑚₁∈𝑀 and all 𝑟^𝑛₂∈𝑅; (2)𝑎𝑟₂⋯(𝑓(𝑠^𝑚₁))⋯𝑟_𝑛=𝑓₀(𝑎𝑟₂⋯𝑠^𝑚₁⋯𝑟_𝑛) for all 𝑎∈𝑀 and all 𝑟^𝑛₂,𝑠^𝑚₁∈𝑅; (3)(𝑎𝑟₂⋯𝑟_𝑛)𝑟_𝑛+1⋯𝑟_2𝑛−1=𝑎𝑟₂⋯𝑔(𝑟^𝑘+𝑛_𝑘+1)⋯𝑟_2𝑛−1 for all 𝑎∈𝑀, 𝑘∈{1,…,𝑛−1}, and 𝑟^𝑘+𝑛_𝑘+1∈𝑅; (4)0_𝑀𝑟₂⋯𝑟_𝑛=0_𝑀 for all 𝑟^𝑛₂∈𝑅; (5)𝑎𝑟₂⋯𝑟_𝑛=0_𝑀 whenever 𝑎∈𝑀, 𝑟^𝑛₂∈𝑅, and 𝑟_𝑖=0 for some 𝑖.

Definition 6. A nonempty subset 𝑁 of a right 𝑛-ary semimodule (𝑀,𝑓₀) over an (𝑚,𝑛)-semiring (𝑅,𝑓,𝑔) is called an 𝑛-ary subsemimodule of 𝑀 if (i) 𝑓₀(𝑎^𝑚₁)∈𝑁 and (ii) 𝑎₁𝑥^𝑛₂∈𝑁 for all 𝑎^𝑚₁∈𝑁 and 𝑥^𝑛₂∈𝑅.
An 𝑛-ary subsemimodule 𝑁 of 𝑀 is called an 𝑛-ary 𝑘-subsemimodule if 𝑓₀(𝑎,𝑏^𝑚₂)∈𝑁;𝑎∈𝑀 and 𝑏^𝑚₂∈𝑁 imply that 𝑎∈𝑁. An 𝑛-ary subsemimodule 𝑁 of 𝑀 is called an 𝑛-ary ℎ-subsemimodule if 𝑓(𝑎,𝑏^𝑚₂,𝑐^𝑚₂)=𝑓(𝑏₁,𝑐^𝑚₂);𝑎,𝑐^𝑚₂∈𝑅 and 𝑏^𝑚₁∈𝑁 imply that 𝑎∈𝑁.

For example, an (𝑚,𝑛)-semiring (𝑅,𝑓,𝑔) can be regarded as a right 𝑛-ary 𝑅-semimodule naturally. Then if 𝐼 is a 𝑘-ideal (an ℎ-ideal) of the (𝑚,𝑛)-semiring (𝑅,𝑓,𝑔), then 𝐼 is also an 𝑛-ary 𝑘-(ℎ-)subsemimodule of this right 𝑛-ary 𝑅-semimodule 𝑅.

Definition 7. A right 𝑛-ary 𝑅-semimodule (𝑀,𝑓₀) is said to be cancellative if (𝑀,𝑓₀) is a cancellative 𝑚-ary semigroup.

Definition 8. An equivalence relation 𝜌 on right 𝑛-ary 𝑅-semimodule (𝑀,𝑓₀) is said to be a congruence relation or simply a congruence of (𝑀,𝑓₀) if the following conditions are satisfied:(1)𝑎^𝑚₁𝜌𝑏^𝑚₁⇒𝑓₀(𝑎^𝑚₁)𝜌𝑓₀(𝑏^𝑚₁) for all 𝑎^𝑚₁,𝑏^𝑚₁∈𝑀, (2)𝑎𝜌𝑏⇒𝑎𝑟^𝑛₂𝜌𝑏𝑟^𝑛₂ for all 𝑎,𝑏∈𝑀 and all 𝑟^𝑛₂∈𝑅.  We say that a congruence 𝜌 of (𝑀,𝑓₀) admits the cancellation law (of addition) if (3)𝑓₀(𝑥,𝑎^𝑚₂)𝜌𝑓₀(𝑦,𝑏^𝑚₂) and 𝑎^𝑚₂𝜌𝑏^𝑚₂ imply 𝑥𝜌𝑦.

Let 𝑁 be an 𝑛-ary subsemimodule of an 𝑛-ary right semimodule (𝑀,𝑓₀) over an (𝑚,𝑛)-semiring 𝑅. Then the congruence on (𝑀,𝑓₀), denoted by 𝑘_𝑁, and defined by setting 𝑥𝑘_𝑁𝑦iff𝑓₀𝑥,𝑎^𝑚₂=𝑓₀𝑦,𝑏^𝑚₂forsome𝑎^𝑚₂,𝑏^𝑚₂∈𝑁, (8) is called the Bourne congruence on 𝑀 defined by the 𝑛-ary subsemimodule 𝑁. We denote the Bourne congruence class of an element 𝑥 by 𝑥/𝑁 and denote the set of all such congruence classes of 𝑀 by 𝑀/𝑁. Define two operations, 𝑚-ary addition and 𝑛-ary scalar multiplication on 𝑀/𝑁, by 𝑓₀(𝑎^𝑚₁/𝑁)=𝑓₀(𝑎^𝑚₁)/𝑁 and (𝑎₁/𝑁)𝑟^𝑛₂=(𝑎₁𝑟^𝑛₂)/𝑁 for all 𝑎^𝑚₁∈𝑀 and all 𝑟^𝑛₂∈𝑅. With these two operations, 𝑀/𝑁 is an 𝑛-ary right semimodule over 𝑅 and we call it the Bourne factor 𝑛-ary semimodule.

Similarly, we can define the Iizuka congruence  ℎ_𝑁 and the Iizuka factor 𝑛-ary semimodule  𝑀[/]𝑁. It is easy to show that 𝑀[/]𝑁 is cancellative.

In what follows, we always assume that the 𝑛-ary right semimodule is cancellative.

3. Primitive (𝑚,𝑛)-Semirings

Definition 9. Let (𝑅,𝑓,𝑔) be an (𝑚,𝑛)-semiring with zero 0. Then the zeroid of 𝑅, denoted by 𝑍(𝑅), is defined as 𝑍(𝑅)=𝑥∈𝑅∶𝑓𝑥,𝑦^𝑚₂=𝑓0,𝑦^𝑚₂forsome𝑦^𝑚₂∈𝑅. (9)

Clearly, the zero element 0 of 𝑅 belongs to 𝑍(𝑅). Furthermore, we have the following.

Lemma 10. The zeroid 𝑍(𝑅) of an (𝑚,𝑛)-semiring (𝑅,𝑓,𝑔) is the smallest ℎ-ideal of (𝑅,𝑓,𝑔).

Proof. It is easily verified that 𝑍(𝑅) is an ideal of 𝑅. To show 𝑍(𝑅) is an ℎ-ideal of 𝑅, we suppose 𝑓(𝑥,𝑦^𝑚₂,𝑧^𝑚₂)=𝑓(𝑦₁,𝑧^𝑚₂), where 𝑥,𝑧^𝑚₂∈𝑅 and 𝑦^𝑚₁∈𝑍(𝑅). For each 𝑖∈{1,…,𝑚} there exist 𝑢(𝑖)^𝑚₂∈𝑅 such that 𝑓(𝑦_𝑖,𝑢(𝑖)^𝑚₂)=𝑓(0,𝑢(𝑖)^𝑚₂), so we have 𝑓𝑓𝑥,𝑦^𝑚₂,𝑧^𝑚₂,𝑢(1)^𝑚₂,…,𝑢(𝑚)^𝑚₂=𝑓𝑓𝑦₁,𝑧^𝑚₂,𝑢(1)^𝑚₂,…,𝑢(𝑚)^𝑚₂, (10) that is, 𝑓𝑥,𝑢(1)^𝑚₂,𝑓𝑦₂,𝑢(2)^𝑚₂,…,𝑓𝑦_𝑚,𝑢(𝑚)^𝑚₂,𝑧^𝑚₂=𝑓𝑓𝑦₁,𝑢(1)^𝑚₂,𝑢(2)^𝑚₂,…,𝑢(𝑚)^𝑚₂,𝑧^𝑚₂. (11)
It follows that 𝑓𝑥,𝑢(1)^𝑚₂,𝑓0,𝑢(2)^𝑚₂,…,𝑓0,𝑢(𝑚)^𝑚₂,𝑧^𝑚₂=𝑓𝑓0,𝑢(1)^𝑚₂,𝑢(2)^𝑚₂,…,𝑢(𝑚)^𝑚₂,𝑧^𝑚₂. (12)
Hence we obtain 𝑓𝑥,𝑢(1)^𝑚₂,𝑢(2)^𝑚₂,…,𝑢(𝑚)^𝑚₂,𝑧^𝑚₂=𝑓0,𝑢(1)^𝑚₂,𝑢(2)^𝑚₂,…,𝑢(𝑚)^𝑚₂,𝑧^𝑚₂, (13) which shows that 𝑥∈𝑍(𝑅), so that 𝑍(𝑅) is an ℎ-ideal of 𝑅.
At last, suppose that 𝐼 is an arbitrary ℎ-ideal of 𝑅. We aim to show 𝑍(𝑅)⊆𝐼. For this, let 𝑥∈𝑍(𝑅). Then there exist 𝑦^𝑚₂∈𝑅 such that 𝑓(𝑥,𝑦^𝑚₂)=𝑓(0,𝑦^𝑚₂), so 𝑓(𝑥,0^𝑚−1,𝑦^𝑚₂)=𝑓(0,𝑦^𝑚₂). It follows that 𝑥∈𝐼 since 𝐼 is an ℎ-ideal and 0∈𝐼. Thus 𝑍(𝑅)⊆𝐼.

Definition 11. Let 𝑀 be a right 𝑛-ary 𝑅-semimodule. The annihilator of 𝑀 in 𝑅, denoted by (0∶𝑀) or 𝐴_𝑅(𝑀), is defined as the subset 𝑟∈𝑅∶𝑎𝑟^𝑚₂=0_𝑀whenever𝑟𝑎∈𝑀,_𝑚2∈𝑅and𝑟_𝑖=𝑟forsome𝑖. (14)

Lemma 12. 𝐴_𝑅(𝑀) is an ℎ-ideal of 𝑅.

Proof. It is obvious that 𝐴_𝑅(𝑀) is an ideal of 𝑅. To show that it is an ℎ-ideal, suppose 𝑓(𝑥,𝑦^𝑚₂,𝑧^𝑚₂)=𝑓(𝑦₁,𝑧^𝑚₂), where 𝑥,𝑧^𝑚₂∈𝑅 and 𝑦^𝑚₁∈𝐴_𝑅(𝑀). Then for all 𝑟_𝑖∈𝑅, 𝑎𝑟₂⋯𝑓𝑥,𝑦^𝑚₂,𝑧^𝑚₂⋯𝑟_𝑛=𝑎𝑟₂⋯𝑓𝑦₁,𝑧^𝑚₂⋯𝑟_𝑛, (15) that is, 𝑓₀𝑎𝑟₂⋯𝑥⋯𝑟_𝑛,𝑎𝑟₂⋯𝑦^𝑚₂⋯𝑟_𝑛,𝑎𝑟₂⋯𝑧^𝑚₂⋯𝑟_𝑛=𝑓₀𝑎𝑟₂⋯𝑦₁⋯𝑟_𝑛,𝑎𝑟₂⋯𝑧^𝑚₂⋯𝑟_𝑛, (16) which deduces that 𝑓₀𝑎𝑟₂⋯𝑥⋯𝑟_𝑛,0^𝑚−1,𝑎𝑟₂⋯𝑧^𝑚₂⋯𝑟_𝑛=𝑓₀0,𝑎𝑟₂⋯𝑧^𝑚₂⋯𝑟_𝑛 (17) since 𝑎𝑟₂⋯𝑦_𝑗⋯𝑟_𝑛=0 for each 𝑗∈{1,…,𝑚}. Thus we have 𝑓₀𝑎𝑟₂⋯𝑥⋯𝑟_𝑛,𝑎𝑟₂⋯𝑧^𝑚₂⋯𝑟_𝑛=𝑓₀0,𝑎𝑟₂⋯𝑧^𝑚₂⋯𝑟_𝑛. (18)
By cancellation law of 𝑀, 𝑎𝑟₂⋯𝑥⋯𝑟_𝑛=0. Hence 𝑥∈𝐴_𝑅(𝑀), as required.

Definition 13. A right 𝑛-ary 𝑅-semimodule 𝑀 is said to be faithful if 𝑍(𝑅)=𝐴_𝑅(𝑀).

One of difficulties when studying the radical of an (𝑚,𝑛)-semiring 𝑅 is how to give an appropriate definition of the irreducibility of a right 𝑛-ary 𝑅-semimodule. The next definition is a generalization of [20, Definition 3.9].

Definition 14. A right 𝑛-ary 𝑅-semimodule 𝑀 is said to be irreducible if for every arbitrary fixed pair 𝑢^𝑚₂,𝑣^𝑚₂∈𝑀 with 𝑢_𝑗≠𝑣_𝑗 for some 𝑗 and for any 𝑥∈𝑀, there exist 𝑎(𝑖)^𝑚₂,𝑏(𝑖)^𝑚₂∈𝑅 with 𝑖=2,…,𝑚 such that 𝑓₀𝑥,𝑢₂𝑎(2)^𝑛₂,…,𝑢_𝑚𝑎(𝑚)^𝑛₂,𝑣₂𝑏(2)^𝑛₂,…,𝑣_𝑚𝑏(𝑚)^𝑛₂=𝑓₀0_𝑀,𝑢₂𝑏(2)^𝑛₂,…,𝑢_𝑚𝑏(𝑚)^𝑛₂,𝑣₂𝑎(2)^𝑛₂,…,𝑣_𝑚𝑎(𝑚)^𝑛₂. (19)

Remark 15. Since 𝑀 is cancellative, it is easily seen that a right 𝑛-ary 𝑅-semimodule 𝑀 is irreducible if and only if for every arbitrary fixed pair 𝑢^𝑚₂,𝑣^𝑚₂∈𝑀 with 𝑢_𝑗≠𝑣_𝑗 for all 𝑗 and for any 𝑥∈𝑀, there exist 𝑎(𝑖)^𝑚₂,𝑏(𝑖)^𝑚₂∈𝑅 with 𝑖=2,…,𝑚 such that equality (2) holds.

Lemma 16. Let 𝐼 be an ℎ-ideal of an (𝑚,𝑛)-semiring 𝑅. If 𝑀 is an irreducible right 𝑛-ary 𝑅/𝐼-semimodule, then 𝑀 is an irreducible right 𝑛-ary 𝑅-semimodule.

Proof. Let 𝑀 be an irreducible right 𝑛-ary 𝑅-semimodule. Then we can define an 𝑛-ary action on 𝑀 by 𝑎𝑟₂⋯𝑟_𝑛=𝑎(𝑟₂/𝐼)⋯(𝑟_𝑛/𝐼) for all 𝑎∈𝑀 and for all 𝑟^𝑛₂∈𝑅, and this makes 𝑀 into an irreducible right 𝑛-ary 𝑅-semimodule.

The converse of Lemma 16 is not necessarily true. But in particular we have the following lemma.

Lemma 17. If 𝑀 is a right 𝑛-ary 𝑅-semimodule then 𝑀 is a right 𝑛-ary 𝑅/𝐴_𝑅(𝑀)-semimodule, where 𝑅/𝐴_𝑅(𝑀) is the Bourne factor semiring. Moreover, if 𝑀 is an irreducible right 𝑛-ary 𝑅-semimodule, then 𝑀 is also an irreducible right 𝑛-ary 𝑅/𝐴_𝑅(𝑀)-semimodule.

Proof. Suppose 𝑀 is a right 𝑛-ary 𝑅-semimodule. We define an 𝑛-ary action on 𝑀 as follows: 𝑎𝑟₂/𝐼⋯𝑟_𝑛/𝐼=𝑎𝑟₂⋯𝑟_𝑛 where 𝐼=𝐴_𝑅(𝑀), for all 𝑎∈𝑀 and for all 𝑟^𝑛₂∈𝑅. We now show that this definition is well-defined. If for each 𝑖=2,…,𝑛, 𝑟_𝑖/𝐼=𝑠_𝑖/𝐼, then 𝑟_𝑖𝑘_𝐼𝑠_𝑖, that is, there exist 𝑥(𝑖)^𝑚₂,𝑦(𝑖)^𝑚₂∈𝐼 such that 𝑓(𝑟_𝑖,𝑥(𝑖)^𝑚₂)=𝑓(𝑠_𝑖,𝑦(𝑖)^𝑚₂). It follows that 𝑎𝑓(𝑟₂,𝑥(2)^𝑚₂)⋯𝑓(𝑟_𝑛,𝑥(𝑛)^𝑚₂)=𝑎𝑓(𝑠₂,𝑦(2)^𝑚₂)⋯𝑓(𝑠_𝑛,𝑦(𝑛)^𝑚₂), which implies that 𝑎𝑟₂⋯𝑟_𝑛=𝑎𝑠₂⋯𝑠_𝑛 since 𝑥(𝑖)^𝑚₂,𝑦(𝑖)^𝑚₂∈𝐴_𝑅(𝑀). Thus 𝑎𝑟₂/𝐼⋯𝑟_𝑛/𝐼=𝑎𝑠₂/𝐼⋯𝑠_𝑛/𝐼, as required. It is easy to see that the above definition makes 𝑀 into a right 𝑛-ary 𝑅-semimodule.
Moreover, if 𝑀 is an irreducible right 𝑛-ary 𝑅-semimodule then it is routine to verify that 𝑀 is also an irreducible right 𝑛-ary 𝑅/𝐴_𝑅(𝑀)-semimodule by (2).

Lemma 18. Let 𝑀 be a right 𝑛-ary 𝑅-semimodule. Then 𝐴_{𝑅/𝐴𝑅(𝑀)}(𝑀)={0/𝐴_𝑅(𝑀)}.

Proof. Let 𝑥/𝐼∈𝐴_𝑅/𝐼(𝑀), where 𝐼=𝐴_𝑅(𝑀). Then for any 𝑎∈𝑀𝑎𝑟₂/𝐼⋯𝑟_𝑛/𝐼=0_𝑀 whenever 𝑟^𝑛₂∈𝑅 and 𝑟_𝑖=𝑥 for some 𝑖∈{2,…,𝑛}. It follows that 𝑎𝑟₂⋯𝑟_𝑛=0_𝑀 where 𝑟_𝑖=𝑥 for some 𝑖∈{2,…,𝑛}. This shows that 𝑥∈𝐼 and so that 𝑥/𝐼=0/𝐼. Consequently, 𝐴_{𝑅/𝐴𝑅(𝑀)}(𝑀)={0/𝐴_𝑅(𝑀)}.

Lemma 19. Any right 𝑛-ary 𝑅-semimodule 𝑀 is a faithful 𝑅/𝐴_𝑅(𝑀)-semimodule.

Proof. Let 𝑀 be a right 𝑛-ary 𝑅-semimodule. Then in view of Lemma 17, 𝑀 is an 𝑅/𝐴_𝑅(𝑀)-semimodule. On the one hand, by Lemma 12, 𝐴_{𝑅/𝐴𝑅(𝑀)}(𝑀) is an ℎ-ideal of 𝑅/𝐴_𝑅(𝑀). On the other hand, by Lemma 10, 𝑍(𝑅/𝐴_𝑅(𝑀)) is the smallest ℎ-ideal of 𝑅/𝐴_𝑅(𝑀). Thus 𝑍(𝑅/𝐴_𝑅(𝑀))⊆𝐴_{𝑅/𝐴𝑅(𝑀)}(𝑀). According to Lemma 18, 𝐴_{𝑅/𝐴𝑅(𝑀)}(𝑀)={0/𝐴_𝑅(𝑀)}. So 𝑍(𝑅/𝐴_𝑅(𝑀))={0/𝐴_𝑅(𝑀)}=𝐴_{𝑅/𝐴𝑅(𝑀)}(𝑀), which means that 𝑀 is a faithful 𝑅/𝐴_𝑅(𝑀)-semimodule.

Lemma 20. If 𝐼 is an ℎ-ideal of an (𝑚,𝑛)-semiring 𝑅 then 𝑍(𝑅/𝐼)={0/𝐼} where 𝑅/𝐼 is the Bourne factor semiring.

Proof. Suppose 𝑥/𝐼∈𝑍(𝑅/𝐼). Then we have 𝑓(𝑥/𝐼,𝑦^𝑚₂/𝐼)=𝑓(0/𝐼,𝑦^𝑚₂/𝐼) for some 𝑦^𝑚₂/𝐼∈𝑅/𝐼. Thus we have 𝑓(𝑥,𝑦^𝑚₂)/𝐼=𝑓(0,𝑦^𝑚₂)/𝐼 which implies that 𝑓(𝑥,𝑦^𝑚₂,𝑎^𝑚₂)=𝑓(0,𝑦^𝑚₂,𝑏^𝑚₂) for some 𝑎^𝑚₂,𝑏^𝑚₂∈𝐼. Hence 𝑓(𝑥,𝑎^𝑚₂,𝑦^𝑚₂)=𝑓(0,𝑏^𝑚₂,𝑦^𝑚₂). This shows that 𝑥∈𝐼 since 𝐼 is an ℎ-ideal of 𝑅. Consequently, 𝑥/𝐼=0/𝐼. Thus 𝑍(𝑅/𝐼)={0/𝐼}.

Definition 21. An (𝑚,𝑛)-semiring 𝑅 is said to be primitive if it has a faithful irreducible cancellative 𝑛-ary 𝑅-semimodule. An ideal 𝑃 is said to be primitive if the Bourne factor semiring 𝑅/𝑃 is primitive.

Evidently, an (𝑚,𝑛)-semiring 𝑅 is primitive if and only if {0} is a primitive ideal of 𝑅. The following theorem characterizes primitive ideals of an (𝑚,𝑛)-semiring.

Theorem 22. An ℎ-ideal 𝑃 of (𝑚,𝑛)-semiring 𝑅 is primitive if and only if 𝑃=𝐴_𝑅(𝑀) for some irreducible right 𝑛-ary 𝑅-semimodule 𝑀.

Proof. Let 𝑃 be an ℎ-ideal of 𝑅 such that 𝑃=𝐴_𝑅(𝑀) for some irreducible right 𝑛-ary 𝑅-semimodule 𝑀. Then by Lemmas 17 and 19  𝑀 is a faithful irreducible 𝑛-ary 𝑅/𝑃-semimodule. This shows that 𝑅/𝑃 is primitive and hence 𝑃 is a primitive ℎ-ideal of 𝑅.
Conversely, let 𝑃 be a primitive ℎ-ideal of 𝑅. Then 𝑅/𝑃 is a primitive (𝑚,𝑛)-semiring. So there exists a faithful irreducible 𝑛-ary 𝑅/𝑃-semimodule 𝑀. Now by Lemma 16  𝑀 is an irreducible 𝑛-ary 𝑅-semimodule. It remains to show that 𝑃=𝐴_𝑅(𝑀). Now 𝑥∈𝐴_𝑅(𝑀)⇔ for all 𝑎∈𝑀 and 𝑟^𝑛₂∈𝑅, 𝑎𝑟₂⋯𝑟_𝑛=0_𝑀 whenever 𝑥=𝑟_𝑖 for some 𝑖∈{2,…,𝑛}⇔𝑎𝑟₂/𝑃⋯𝑟_𝑛/𝑃=0_𝑀/𝑃 whenever 𝑥/𝑃=𝑟_𝑖/𝑃 for some 𝑖∈{2,…,𝑛}⇔𝑥/𝑃∈𝐴_𝑅/𝑃(𝑀)=𝑍(𝑅/𝑃) since 𝑀 is a faithful 𝑛-ary 𝑅/𝑃-semimodule ⇔𝑥/𝑃∈𝐴_𝑅/𝑃(𝑀)={0/𝑃}, by Lemma 20  ⇔𝑥/𝑃=0/𝑃⇔𝑥∈𝑃. Thus 𝑃=𝐴_𝑅(𝑀) as desired.

4. Jacobson Radical of an (𝑚,𝑛)-Semiring

Let us begin this section by defining the semi-irreducibility of a right 𝑛-ary 𝑅-semimodule.

Definition 23. A right 𝑛-ary 𝑅-semimodule 𝑀 is said to be semi-irreducible if 𝑀𝑅^𝑛−1≠{0_𝑀}; that is, 𝑎𝑟₂⋯𝑟_𝑛≠0_𝑀 for some 𝑎∈𝑀 and some 𝑟^𝑛₂∈𝑅, and 𝑀 does not contain any 𝑛-ary 𝑘-subsemimodule other than {0_𝑀} and 𝑀.

Lemma 24. Let 𝐼 be a subset of an (𝑚,𝑛)-semiring 𝑅 and 𝑀 a right 𝑛-ary 𝑅-semimodule with 𝑀𝑅^𝑖−2𝐼𝑅^𝑛−𝑖≠{0_𝑀} for some 𝑖∈{2,…,𝑛}. In the case where 𝑖=2, we assume further that 𝐼 is a left ideal of 𝑅. Then the following statements are true:(1)If 𝑀 is semi-irreducible and 𝑎∈𝑀, then 𝑎=0 if and only if 𝑎𝑅^𝑖−2𝐼𝑅^𝑛−𝑖={0_𝑀}; (2)If 𝑀 is irreducible and 𝑎,𝑏∈𝑀, then 𝑎=𝑏 if and only if 𝑎𝑟₂^𝑖−1𝑠𝑟^𝑛_𝑖+1=𝑏𝑟₂^𝑖−1𝑠𝑟^𝑛_𝑖+1 for all 𝑟_𝑗∈𝑅 and all 𝑠∈𝐼.

Proof. Suppose that (𝑀,𝑓₀) is a semi-irreducible right 𝑛-ary semimodule over an (𝑚,𝑛)-semiring (𝑅,𝑓,𝑔), and that 𝐼 is a subset of 𝑅 such that 𝑀𝑅^𝑖−2𝐼𝑅^𝑛−𝑖≠{0_𝑀} for some 𝑖∈{2,…,𝑛}. In the case where 𝑖=2, we further assume that 𝐼 is a left ideal of 𝑅.
(1) Assume that 𝑀 is semi-irreducible. Let 𝑎∈𝑀 be such that 𝑎𝑅^𝑖−2𝐼𝑅^𝑛−𝑖=0_𝑀. (20)
Set 𝑀₀=𝑥∈𝑀∶𝑥𝑅^𝑖−2𝐼𝑅^𝑛−𝑖=0_𝑀. (21)
It is clear that 𝑎∈𝑀₀, and it is easy to show that 𝑀₀ is a subsemimodule of 𝑀. Let 𝑓₀(𝑥,𝑦^𝑚₂)∈𝑀₀ and 𝑦^𝑚₂∈𝑀₀. Then 𝑓₀(𝑥,𝑦^𝑚₂)𝑅^𝑖−2𝐼𝑅^𝑛−𝑖={0_𝑀} and 𝑦^𝑚₂𝑅^𝑖−2𝐼𝑅^𝑛−𝑖={0_𝑀}. Thus 𝑥𝑅^𝑖−2𝐼𝑅^𝑛−𝑖={0_𝑀}; that is, 𝑥∈𝑀₀. This shows that 𝑀₀ is a 𝑘-subsemimodule of 𝑀. Since 𝑀𝑅^𝑖−2𝐼𝑅^𝑛−𝑖≠{0_𝑀}, 𝑀₀≠𝑀. Since 𝑀 is semi-irreducible, 𝑀₀={0_𝑀} and therefore 𝑎=0.
The converse part is obvious.
(2) Assume that 𝑀 is irreducible. Let 𝑎,𝑏∈𝑀 be such that 𝑎≠𝑏. Set 𝑢_𝑗=𝑎, 𝑣_𝑗=𝑏 for 𝑗=2,…,𝑚. Since 𝑀𝑅^𝑖−2𝐼𝑅^𝑛−𝑖≠{0_𝑀}, we have 𝑥𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1≠0_𝑀 for some 𝑥∈𝑀,𝑠∈𝐼 and 𝑟_𝑗∈𝑅. Since 𝑀 is irreducible, according to Definition 14, there exist 𝑎(𝑖)^𝑚₂,𝑏(𝑖)^𝑚₂∈𝑅 with 𝑖=2,…,𝑚 such that 𝑓₀𝑥,𝑢₂𝑎(2)^𝑛₂,…,𝑢_𝑚𝑎(𝑚)^𝑛₂,𝑣₂𝑏(2)^𝑛₂,…,𝑣_𝑚𝑏(𝑚)^𝑛₂=𝑓₀0_𝑀,𝑢₂𝑏(2)^𝑛₂,…,𝑢_𝑚𝑏(𝑚)^𝑛₂,𝑣₂𝑎(2)^𝑛₂,…,𝑣_𝑚𝑎(𝑚)^𝑛₂. (22)
Hence 𝑓₀𝑥𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1,𝑢₂𝑎(2)^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1,…,𝑢_𝑚𝑎(𝑚)^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1,𝑣₂𝑏(2)^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1,…,𝑣_𝑚𝑏(𝑚)^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1=𝑓₀0_𝑀𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1,𝑢₂𝑏(2)^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1𝑢,…,_𝑚𝑏(𝑚)^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1,𝑣₂𝑎(2)^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1,…,𝑣_𝑚𝑎(𝑚)^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1=𝑓₀0_𝑀,𝑢₂𝑏(2)^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1,…,𝑢_𝑚𝑏(𝑚)^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1,𝑣₂𝑎(2)^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1,…,𝑣_𝑚𝑎(𝑚)^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1. (23)
Since 𝑀 is cancellative and 𝑥𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1≠0_𝑀, at least one of the following 2(𝑚−1) equalities does not hold: 𝑢_𝑗𝑎𝑗^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1=𝑣_𝑗𝑎(2)^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1where𝑢𝑗=2,…,𝑚;_𝑗𝑏𝑗^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1=𝑣_𝑗𝑏(2)^𝑛₂𝑟₂^𝑖−1𝑠𝑟^𝑛−𝑖_𝑖+1,where𝑗=2,…,𝑚. (24)
So we conclude that if 𝑎𝑟₂^𝑖−1𝑠𝑟^𝑛_𝑖+1=𝑏𝑟₂^𝑖−1𝑠𝑟^𝑛_𝑖+1 for all 𝑟_𝑗∈𝑅 and all 𝑠∈𝐼, then 𝑎=𝑏.
The converse part follows easily.

Lemma 25. Let 𝑀 be a right 𝑛-ary 𝑅-semimodule and 𝑀≠{0_𝑀}. Then 𝑀 is semi-irreducible if and only if for every nonzero 𝑎∈𝑀, 𝑎𝑅^𝑛−1=𝑀.

Proof. Assume that 𝑀 is a semi-irreducible right 𝑛-ary 𝑅-semimodule and 𝑀≠{0_𝑀}. Let 𝑎∈𝑀 be such that 𝑎≠0_𝑀. Then by Lemma 24  𝑎𝑅^𝑛−1≠{0_𝑀}. Since 𝑎𝑅^𝑛−1 is an 𝑛-ary 𝑘-subsemimodule of 𝑀, 𝑎𝑅^𝑛−1=𝑀.
Conversely, suppose that for any nonzero 𝑎∈𝑀, 𝑎𝑅^𝑛−1=𝑀. Let 𝑁≠{0_𝑀} be an 𝑛-ary 𝑘-subsemimodule of 𝑀. Then there exists 𝑏∈𝑁 such that 𝑏≠0_𝑀. So by hypothesis, 𝑏𝑅^𝑛−1=𝑀. Hence for any 𝑥∈𝑀, there exist 𝑎^𝑚₁∈𝑏𝑅^𝑛−1 such that 𝑓₀(𝑥,𝑎^𝑚₂)=𝑎₁. Since 𝑏𝑅^𝑛−1⊆𝑁, we have 𝑎^𝑚₁∈𝑁. Since 𝑁 is an 𝑛-ary 𝑘-subsemimodule, 𝑓₀(𝑥,𝑎^𝑚₂)=𝑎₁ implies that 𝑥∈𝑁. This shows that 𝑁=𝑀. Now if 𝑀𝑅^𝑛−1={0_𝑀} then 𝑎𝑅^𝑛−1={0_𝑀} for all 𝑎∈𝑀. Hence 𝑎𝑅^𝑛−1={0_𝑀}. So we have 𝑀={0_𝑀}, a contradiction. Therefore, 𝑀𝑅^𝑛−1≠{0_𝑀}. Thus 𝑀 is semi-irreducible.

Corollary 26. If a right 𝑛-ary 𝑅-semimodule 𝑀 is irreducible, then it is semi-irreducible and 𝑀𝑅^𝑛−1=𝑀.

Proof. Assume that 𝑀 is an irreducible right 𝑛-ary 𝑅-semimodule. Then 𝑀≠{0_𝑀} and, consequently, there exists a nonzero 𝑦∈𝑀. In view of (2) with 𝑢_𝑗=𝑦 and 𝑣_𝑗=0_𝑀 for 𝑗=2,…,𝑚, we obtain that for any 𝑥∈𝑀 there exist 𝑎(𝑗)^𝑛₂∈𝑅 with 𝑗=2,…,𝑚 such that 𝑓₀𝑥,𝑦𝑎(2)^𝑛₂,…,𝑦𝑎(𝑚)^𝑛₂,0^()𝑚−1_𝑀=𝑓₀0_𝑀,𝑦𝑏(2)^𝑛₂,…,𝑦𝑏(𝑚)^𝑛₂,0^()𝑚−1_𝑀, (25) so that 𝑓₀𝑥,𝑦𝑎(2)^𝑛₂,…,𝑦𝑎(𝑚)^𝑛₂=𝑓₀0_𝑀,𝑦𝑏(2)^𝑛₂,…,𝑦𝑏(𝑚)^𝑛₂∈𝑦𝑅^𝑛−1. (26)
It follows that 𝑥∈𝑦𝑅^𝑛−1. Thus 𝑦𝑅^𝑛−1=𝑀. By Lemma 25, 𝑀 is semi-irreducible.
Furthermore, 𝑀𝑅^𝑛−1≠{0_𝑀}, which implies that 𝑀𝑅^𝑛−1≠{0_𝑀}. Since 𝑀𝑅^𝑛−1 is an 𝑛-ary 𝑘-subsemimodule of 𝑀, 𝑀𝑅^𝑛−1=𝑀 as required.

Now we can define the Jacobson radical of an (𝑚,𝑛)-semiring in an external way.

Definition 27. Let 𝑅 be an (𝑚,𝑛)-semiring and Δ be the set of all irreducible right 𝑛-ary 𝑅-semimodules. Then 𝐽(𝑅)=⋂_𝑀∈Δ𝐴_𝑅(𝑀) is called the Jacobson radical of 𝑅. If Δ is empty then 𝑅 itself is considered as 𝐽(𝑅); that is, 𝐽(𝑅)=𝑅, and in this case, we say that 𝑅 is a radical (𝑚,𝑛)-semiring. An (𝑚,𝑛)-semiring 𝑅 is said to be Jacobson semisimple or J-semisimple if 𝐽(𝑅)={0}.

By Lemma 12, 𝐴_𝑅(𝑀) is an ℎ-ideal of 𝑅. Note that the intersection of any family of ℎ-ideals is again an ℎ-ideal. Consequently, we obtain the following.

Lemma 28. 𝐽(𝑅) is an ℎ-ideal of 𝑅.

Lemma 29. If 𝑀 is a right 𝑛-ary 𝑅-semimodule then 𝑀 is a right 𝑛-ary 𝑅/𝐽(𝑅)-semimodule, where 𝑅/𝐽(𝑅) is the Bourne factor semiring. Moreover, if 𝑀 is an irreducible right 𝑛-ary 𝑅-semimodule, then 𝑀 is also an irreducible right 𝑛-ary 𝑅/𝐽(𝑅)-semimodule.

Proof. This lemma can be proved by the same method as in proving Lemma 17.

Theorem 30. If 𝑅 is an (𝑚,𝑛)-semiring, then the Bourne factor semiring 𝑅/𝐽(𝑅) is Jacobson semisimple.

Proof. By Δ and Λ, we denote the set of all irreducible right 𝑛-ary 𝑅-semimodules and the set of all irreducible right 𝑛-ary 𝑅/𝐽(𝑅)-semimodules, respectively. Then according to Lemmas 28, 16, and, 29, we obtain that Δ=Λ. For any 𝑥∈𝐽(𝑅/𝐽(𝑅)) and any 𝑀∈Δ, we have 𝑥∈𝐴_{𝑅/𝐽(𝑅)}(𝑀), which means that for any 𝑎∈𝑀, 𝑎𝑟₂/𝐽(𝑅)⋯𝑟_𝑛/𝐽(𝑅)=0_𝑀 whenever 𝑟^𝑛₂∈𝑅 and 𝑟_𝑖=𝑥 for some 𝑖∈{2,…,𝑛}. Thus 𝑎𝑟₂⋯𝑟_𝑛=0_𝑀 whenever 𝑟^𝑛₂∈𝑅 and 𝑟_𝑖=𝑥 for some 𝑖∈{2,…,𝑛}, so 𝑥∈𝐴_𝑅(𝑀) for all 𝑀∈Λ. That is, 𝑥∈𝐽(𝑅). Hence 𝑥/𝐽(𝑅)=0/𝐽(𝑅). We have shown that 𝐽(𝑅/𝐽(𝑅))={0/𝐽(𝑅)}. By Definition 27, 𝑅/𝐽(𝑅) is Jacobson semisimple.

The next theorem is a direct corollary of Theorem 22, giving an internal characterization of the Jacobson radical of an (𝑚,𝑛)-semiring.

Theorem 31. 𝐽(𝑅) is the intersection of all primitive ℎ-ideals of 𝑅.

Definition 32. Let 𝑃 be an 𝑖-ideal of an (𝑚,𝑛)-semiring 𝑅 for some 𝑖∈{1,…,𝑛}. Then 𝑃 is said to be strongly seminilpotent if there exists a positive integer 𝑡 such that (𝑃𝑅^𝑛−2)^𝑡−1𝑃⊆𝑍(𝑅), where 𝑅^𝑛−2=𝑅⋯^𝑛−2𝑅, (𝑃𝑅^𝑛−2)^𝑡−1=(𝑃𝑅^𝑛−2)(𝑃𝑅^𝑛−2)⋯𝑡−1 times, (𝑃𝑅^𝑛−1)⁰𝑃=𝑃. 𝑃 is said to be strongly nilpotent if there exists a positive integer 𝑡 such that (𝑃𝑅^𝑛−2)^𝑡−1𝑃={0}.

Theorem 33. If 𝑃 is a strongly semi-nilpotent left ideal of 𝑅, then 𝑃⊆𝐽(𝑅).

Proof. Suppose on the contrary that 𝑃⊆𝐽̸(𝑅)=_𝑀∈Δ𝐴_𝑅(𝑀), (27) where 𝑅 is an (𝑚,𝑛)-semiring and Δ is the set of all irreducible right 𝑛-ary 𝑅-semimodules. Then there exists an 𝑀∈Δ such that 𝑃⊆𝐴̸_𝑅(𝑀). Thus there exists 𝑖∈{2,…,𝑛} such that 𝑀𝑅^𝑖−2𝑃𝑅^𝑛−𝑖≠0_𝑀. (28)
Since 𝑃 is strongly semi-nilpotent, there exists a positive integer 𝑡 such that (𝑃𝑅^𝑛−2)^𝑡−1𝑃⊆𝑍(𝑅). By Lemmas 10 and 12, 𝑍(𝑅)⊆𝐴_𝑅(𝑀). It follows that (𝑃𝑅^𝑛−2)^𝑡−1𝑃⊆𝐴_𝑅(𝑀), which implies that 𝑀𝑅^𝑖−2𝑃𝑅^{𝑛−2𝑡−1}𝑃𝑅^𝑛−𝑖=0_𝑀. (29) If (29) holds for all positive integers 𝑡’s, then in particular it is true for 𝑡=1 and in this case we have 𝑀𝑅^𝑖−2𝑃𝑅^𝑛−𝑖={0_𝑀}, a contradiction to (28). If (29) does not hold for all 𝑡, then there exist 𝑥∈𝑀 and positive 𝑠 such that 𝑥𝑅^𝑖−2𝑃𝑅^{𝑛−2𝑠−1}𝑃𝑅^𝑛−1≠0_𝑀,𝑥𝑅^𝑖−2𝑃𝑅^𝑛−2𝑠𝑃𝑅^𝑛−𝑖=0_𝑀. (30) Thus 𝑥≠0_𝑀 and there exists 𝑎∈𝑥𝑅^𝑖−2(𝑃𝑅^𝑛−2)^𝑠−1𝑃𝑅^𝑛−𝑖 such that 𝑎≠0_𝑀. It follows that 𝑎𝑅^𝑖−2𝑃𝑅^𝑛−𝑖⊆𝑥𝑅^𝑖−2𝑃𝑅^{𝑛−2𝑠−1}𝑃𝑅^𝑛−𝑖𝑅^𝑖−2𝑃𝑅^𝑛−𝑖=𝑥𝑅^𝑖−2𝑃𝑅^𝑛−2𝑠𝑃𝑅^𝑛−𝑖=0_𝑀, (31)
so we have 𝑎𝑅^𝑖−2𝑃𝑅^𝑛−𝑖=0_𝑀. (32)
By Lemma 24, we obtain 𝑎=0_𝑀, again a contradiction. This completes the proof.

The next result is a direct corollary of Theorem 33.

Corollary 34. If an (𝑚,𝑛)-semiring 𝑅 is Jacobson semisimple then 𝑅 does not contain any non-zero strongly semi-nilpotent left ideal and hence 𝑅 does not contain any nontrivial strongly nilpotent left ideal.

Lemma 35. If 𝑀 is a (semi-)irreducible right 𝑛-ary 𝑅-semimodule and 𝑁≠{0_𝑀} is an arbitrary 𝑅-subsemimodule (and 𝑁𝑅^𝑛−1≠{0_𝑀}), then 𝑁 is (semi-)irreducible, and for any 𝑥^𝑛₂,𝑦^𝑛₂∈𝑅 the following statement is true: the equality 𝑢𝑥^𝑛₂=𝑢𝑦^𝑛₂ holds for all 𝑢∈𝑀 if and only if the same equality holds for all 𝑢∈𝑁. Furthermore, 𝐴_𝑅(𝑀)=𝐴_𝑅(𝑁).

Proof. Assume 𝑀 is an irreducible right 𝑛-ary 𝑅-semimodule. Then from (2), it follows that 𝑁 is irreducible. If 𝑀 is a semi-irreducible and 𝑁𝑅^𝑛−1≠{0_𝑀}, then 𝑁 is semi-irreducible by Definition 23 since any subsemimodule of 𝑁 is clearly a subsemimodule of 𝑀.
Let 𝑥^𝑛₂,𝑦^𝑛₂∈𝑅 be such that the equality 𝑢𝑥^𝑛₂=𝑢𝑦^𝑛₂ holds for all 𝑢∈𝑀. Since 𝑀 is semi-irreducible, for any 𝑎(≠0_𝑀)∈𝑀 and any 𝑏(≠0_𝑀)∈𝑁, there exist positive 𝑝, 𝑞 and 𝑥(𝑖)^𝑝₂,𝑦(𝑗)^𝑞₂∈𝑅 such that 𝑝≡𝑞≡0mod(𝑚−1) and 𝑓₀𝑎,𝑏𝑥(2)^𝑛₂,…,𝑏𝑥𝑝^𝑛₂=𝑓₀0_𝑀,𝑏𝑦(2)^𝑛₂,…,𝑏𝑦𝑞^𝑛₂. (33)
Thus we have the following two equalities: 𝑓₀𝑎𝑥^𝑛₂,𝑏𝑥(2)^𝑛₂𝑥^𝑛₂,…,𝑏𝑥𝑝^𝑛₂𝑥^𝑛₂=𝑓₀0_𝑀,𝑏𝑦(2)^𝑛₂𝑥^𝑛₂,…,𝑏𝑦𝑞^𝑛₂𝑥^𝑛₂,𝑓₀𝑎𝑦^𝑛₂,𝑏𝑥(2)^𝑛₂𝑦^𝑛₂,…,𝑏𝑥𝑝^𝑛₂𝑦^𝑛₂=𝑓₀0_𝑀,𝑏𝑦(2)^𝑛₂𝑦^𝑛₂,…,𝑏𝑦𝑞^𝑛₂𝑦^𝑛₂. (34)
It follows that 𝑓₀0_𝑀,𝑎𝑥^𝑛₂,𝑏𝑥(2)^𝑛₂𝑥^𝑛₂,…,𝑏𝑥𝑝^𝑛₂𝑥^𝑛₂,𝑏𝑦(2)^𝑛₂𝑦^𝑛₂,…,𝑏𝑦𝑞^𝑛₂𝑦^𝑛₂=𝑓₀0_𝑀,𝑎𝑦^𝑛₂,𝑏𝑥(2)^𝑛₂𝑦^𝑛₂,…,𝑏𝑥𝑝^𝑛₂𝑦^𝑛₂,𝑏𝑦(2)^𝑛₂𝑥^𝑛₂,…,𝑏𝑦𝑞^𝑛₂𝑥^𝑛₂. (35)
Observing that 𝑏𝑥(𝑖)^𝑛₂,𝑏𝑦(𝑗)^𝑛₂∈𝑁, since 𝑁 is a submodule, we have 𝑏𝑥(𝑖)^𝑛₂𝑥^𝑛₂=𝑏𝑥(𝑖)^𝑛₂𝑦^𝑛₂ and 𝑏𝑦(2)^𝑛₂𝑥^𝑛₂=𝑏𝑦(2)^𝑛₂𝑦^𝑛₂ for all 𝑖,𝑗 by the assumption. Hence by cancellation law, (35) deduces that 𝑎𝑥^𝑛₂=𝑎𝑦^𝑛₂. The converse implication is clear.
Furthermore, letting 𝑦_𝑖=0 for some 𝑖, we get that the equality 𝑢𝑥^𝑛₂=0_𝑀 holds for all 𝑢∈𝑀 if and only if the same equality holds for all 𝑢∈𝑁. Thus 𝐴_𝑅(𝑀)=𝐴_𝑅(𝑁).

Lemma 36. Let 𝐼 be an ideal of an (𝑚,𝑛)-semiring 𝑅. (1)If 𝑀 is an (semi-)irreducible right 𝑛-ary 𝑅-semimodule (and 𝑀𝑅^𝑛−1≠{0_𝑀}), then 𝑀 is an (semi-)irreducible right 𝑛-ary 𝐼-semimodule. (2)If 𝑀 is an irreducible right 𝑛-ary 𝐼-semimodule, then there exists an irreducible right 𝑛-ary 𝑅-semimodule 𝑀^∗, which can be regarded as an 𝐼-subsemimodule 𝑁 of 𝑀.

Proof. (1) Let 𝑀 be an irreducible 𝑅-semimodule and 𝑢^𝑚₂,𝑣^𝑚₂∈𝑀 be such that 𝑢_𝑗≠𝑣_𝑗 for some 𝑗. Without loss of generality, we suppose that 𝑀≠{0_𝑀}. From (2) we deduce that 𝑀𝑅^𝑛−1≠{0_𝑀}. By Lemma 24, 𝑢_𝑗𝑐^𝑛₂≠𝑣_𝑗𝑐^𝑛₂ for some 𝑐^𝑛₂∈𝐼. Since 𝑀 is irreducible, by (19) there exist 𝑎(𝑖)^𝑚₂,𝑏(𝑖)^𝑚₂∈𝑅 with 𝑖=2,…,𝑚 such that 𝑓₀𝑥,𝑢₂𝑐^𝑛₂𝑎(2)^𝑛₂,…,𝑢_𝑚𝑐^𝑛₂𝑎(𝑚)^𝑛₂,𝑣₂𝑐^𝑛₂𝑏(2)^𝑛₂,…,𝑣_𝑚𝑐^𝑛₂𝑏(𝑚)^𝑛₂=𝑓₀0_𝑀,𝑢₂𝑐^𝑛₂𝑏(2)^𝑛₂,…,𝑢_𝑚𝑐^𝑛₂𝑏(𝑚)^𝑛₂,𝑣₂𝑐^𝑛₂𝑎(2)^𝑛₂,…,𝑣_𝑚𝑐^𝑛₂𝑎(𝑚)^𝑛₂, (36) that is, 𝑓₀𝑥,𝑢₂𝑐^𝑛₂𝑎(2)^𝑛₂,…,𝑢_𝑚𝑐^𝑛₂𝑎(𝑚)^𝑛₂,𝑣₂𝑐^𝑛₂𝑏(2)^𝑛₂,…,𝑣_𝑚𝑐^𝑛₂𝑏(𝑚)^𝑛₂=𝑓₀0_𝑀,𝑢₂𝑐^𝑛₂𝑏(2)^𝑛₂,…,𝑢_𝑚𝑐^𝑛₂𝑏(𝑚)^𝑛₂,𝑣₂𝑐^𝑛₂𝑎(2)^𝑛₂,…,𝑣_𝑚𝑐^𝑛₂𝑎(𝑚)^𝑛₂, (37) which means that 𝑀 is an irreducible 𝐼-semimodule by (19) again since for all 𝑗∈{2,…,𝑛}𝑐^𝑛₂𝑎(𝑗)^𝑛₂,𝑐^𝑛₂𝑏(𝑗)^𝑛₂∈𝐼^𝑛−1.
Assume that 𝑀 is a semi-irreducible 𝑅-semimodule and 𝑀𝐼^𝑛−1≠{0_𝑀}. According to Lemma 24, for any 𝑢(≠0_𝑀)∈𝑀 there exist 𝑏^𝑛₂∈𝑅 such that 𝑢𝑏^𝑛₂≠0_𝑀. By Lemma 25, (𝑢𝑏^𝑛₂)𝑅^𝑛−1=𝑀, so for any 𝑥∈𝑀 there exist positive integers 𝑝,𝑞 and 𝑥(𝑖)^𝑛₂,𝑦(𝑗)^𝑛₂∈𝑅 such that 𝑝≡𝑞≡0mod(𝑚−1) and 𝑓₀𝑥,𝑢𝑏^𝑛₂𝑥(2)^𝑛₂,…,𝑢𝑏^𝑛₂𝑥𝑝^𝑛₂=𝑓₀0_𝑀,𝑢𝑏^𝑛₂𝑦(2)^𝑛₂,…,𝑢𝑏^𝑛₂𝑦𝑞^𝑛₂, (38) which shows that 𝑓₀𝑥,𝑢𝑏^𝑛₂𝑥(2)^𝑛₂,…,𝑢𝑏^𝑛₂𝑥𝑝^𝑛₂=𝑓₀0_𝑀,𝑢𝑏^𝑛₂𝑥(2)^𝑛₂,…,𝑢𝑏^𝑛₂𝑥𝑞^𝑛₂. (39)
Note that for all 𝑖, 𝑗, (𝑏^𝑛₂𝑥(𝑖)^𝑛₂),(𝑏^𝑛₂𝑥(𝑗)^𝑛₂)∈𝐼^𝑛−1. Thus we obtain 𝑢𝐼^𝑛−1=𝑀. By Lemma 25 again, 𝑀 is a semi-irreducible right 𝑛-ary 𝐼-semimodule.
(2) Let 𝑀 be an irreducible right 𝑛-ary 𝐼-semimodule, and let 𝑁=𝑀𝐼^𝑛−1. Then 𝑁≠{0_𝑀} and 𝑁 is an 𝐼-subsemimodule of 𝑀. Thus by Lemma 35, 𝑁 is irreducible and for any 𝑥^𝑛₂,𝑦^𝑛₂∈𝑅 the following conclusion is true: the equality 𝑢𝑥^𝑛₂=𝑢𝑦^𝑛₂ holds for all 𝑢∈𝑀 if and only if the same equality holds for all 𝑢∈𝑁. If 𝑓₀(𝑎(1)𝑥(1)^𝑛₂,…,𝑎(𝑝)𝑥(𝑝)^𝑛₂)=𝑓₀(𝑏(1)𝑦(1)^𝑛₂,…,𝑏(𝑞)𝑦(𝑞)^𝑛₂) for some 𝑥(𝑖)^𝑛₂,𝑦(𝑗)^𝑛₂∈𝐼 and 𝑎(𝑖),𝑏(𝑗)∈𝑁, then for any 𝑟^𝑛₂∈𝑅 and any 𝑧^𝑛₂∈𝐼, 𝑓₀𝑎(1)𝑥(1)^𝑛₂𝑟^𝑛₂,…,𝑎𝑝𝑥𝑝^𝑛₂𝑟^𝑛₂𝑧^𝑛₂=𝑓₀𝑎(1)𝑥(1)^𝑛₂𝑟^𝑛₂𝑧^𝑛₂,…,𝑎𝑝𝑥𝑝^𝑛₂𝑟^𝑛₂𝑧^𝑛₂=𝑓₀𝑎(1)𝑥(1)^𝑛₂,…,𝑎𝑝𝑥𝑝^𝑛₂𝑟^𝑛₂𝑧^𝑛₂=𝑓₀𝑏(1)𝑦(1)^𝑛₂,…,𝑏𝑞𝑦𝑞^𝑛₂𝑟^𝑛₂𝑧^𝑛₂=𝑓₀𝑏(1)𝑦(1)^𝑛₂𝑟^𝑛₂𝑧^𝑛₂,…,𝑏𝑞𝑦𝑞^𝑛₂𝑟^𝑛₂𝑧^𝑛₂=𝑓₀𝑏(1)𝑦(1)^𝑛₂𝑟^𝑛₂,…,𝑏𝑞𝑦𝑞^𝑛₂𝑟^𝑛₂𝑧^𝑛₂, (40)which implies that 𝑓₀𝑎(1)𝑥(1)^𝑛₂𝑟^𝑛₂,…,𝑎𝑝𝑥𝑝^𝑛₂𝑟^𝑛₂=𝑓₀𝑏(1)𝑦(1)^𝑛₂𝑟^𝑛₂,…,𝑏𝑞𝑦𝑞^𝑛₂𝑟^𝑛₂ (41) by Lemma 24 since 𝑀 is an irreducible right 𝑛-ary 𝐼-semimodule. Thus we can define an operation on 𝑁𝑅^𝑛−1 into 𝑁 by setting 𝑓₀𝑎(1)𝑥(1)^𝑛₂,…,𝑎𝑝𝑥𝑝^𝑛₂𝑟^𝑛₂=𝑓₀𝑎(1)𝑥(1)^𝑛₂𝑟^𝑛₂,…,𝑎𝑝𝑥𝑝^𝑛₂𝑟^𝑛₂, (42) where 𝑥(𝑖)^𝑛₂,𝑦(𝑗)^𝑛₂∈𝐼 and 𝑎(𝑖),𝑏(𝑗)∈𝑁. Thus 𝑁 with the addition and the above operation becomes a right 𝑛-ary 𝑅-semimodule 𝑀^∗ which, as a right 𝑛-ary 𝐼-semimodule, is isomorphic to the right 𝑛-ary 𝐼-semimodule 𝑁. It is clear that 𝑀^∗ is an irreducible right 𝑛-ary 𝑅-semimodule.

Now we are ready to generalize [17, Theorem 2].

Theorem 37. If 𝐼 is an ideal of an (𝑚,𝑛)-semiring 𝑅, then 𝐽(𝐼)=𝐽(𝑅)∩𝐼.

Proof. Let 𝑅 be an (𝑚,𝑛)-semiring and let Δ be the set of all irreducible right 𝑛-ary 𝑅-semimodules. Then by Definition 27  𝐽(𝑅)=⋂_𝑀∈Δ𝐴_𝑅(𝑀). If 𝐼 is an ideal of an (𝑚,𝑛)-semiring 𝑅, then 𝐽(𝐼)=⋂_𝑀∈Λ𝐴_𝐼(𝑀), where Λ is the set of all irreducible right 𝑛-ary 𝐼-semimodules.
For any 𝑀∈Δ, according to Lemma 36, we have 𝑀∈Λ. It is evident that 𝐴_𝐼(𝑀)=𝐴_𝑅(𝑀)∩𝐼. This shows that 𝐽(𝐼)⊆𝐽(𝑅)∩𝐼.
For any 𝑀∈Λ, according to Lemma 36, we have that 𝑀^∗∈Δ and that 𝑀^∗ can be regarded as an 𝐼-subsemimodule 𝑁 of 𝑀. By Lemma 35, we have that 𝐴_𝐼(𝑀)=𝐴_𝐼(𝑁)=𝐴_𝐼(𝑀^∗)=𝐴_𝑅(𝑀^∗)∩𝐼. This shows that 𝐽(𝐼)⊇𝐽(𝑅)∩𝐼.
Summarizing the above, we obtain that 𝐽(𝐼)=𝐽(𝑅)∩𝐼.

Consequently, we have.

Theorem 38. For an (𝑚,𝑛)-semiring 𝑅, 𝐽(𝑅) is a radical (𝑚,𝑛)-semiring; that is, 𝐽(𝐽(𝑅))=𝐽(𝑅).

Acknowledgments

This work was supported by the National Natural Foundation of China (no. 10571005) and the Shandong Province Natural Science Foundation ZR2011AM005.