#### Abstract

The notion of -ary semimodules is introduced so that the Jacobson radical of an -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 -semiring.

#### 1. Introduction

The concept of semigroups [1] was generalized to that of ternary semigroups [2], that of -ary semigroups [3–6], and even to that of -semigroups [7]. Similarly, it was natural to generalize the notion of rings to that of ternary semirings, that of -ary semirings, and even that of -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 -semirings [21], even to -semirings [22–24]. The radicals of ternary semirings and of -semirings were studied in [20, 21], respectively. The concept of -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 , 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 -semirings by means of -ary semimodules so that we can define externally the Jacobson radical of an -semiring, and then we establish the radical properties of the Jacobson radical of an -semiring. Some necessary notions such as irreducible -ary semimodules over an -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 is denoted by . Thus the following expression is represented as In the case when , then (2) is expressed as If , then (2) can be written as .

Recall that an -ary semigroup is defined as a nonempty set with an -ary associative operation ; that is, for all and all . Whence we may denote by briefly. Generally, we have the notation for each positive and all . Thus for positive integer , is well defined if and only if ; see [7, Lemma 1.1]. An -ary semigroup is called cancellative if for all .

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

*Definition 1. *A nonempty set together with an -ary operation , called *addition,* and an -ary operation , called *multiplication*, is said to be an *-semiring* if the following conditions are satisfied. (1) is an -ary semigroup and is an -ary semigroup.(2) is distributive with respect to operation ; that is, for every , , ,
(3) is commutative; that is, for every permutation of and all ,
(4) There is an element , called the *zero* of , satisfying the following two properties: (4A) is an -identity; that is, for every ;(4B) is a -zero; that is, for all , whenever there exists such that .

It is clear that the zero of an -semiring is necessarily unique.

*Definition 2. * An -semiring is called *additively cancellative* if the -ary semigroup is cancellative and *multiplicatively cancellative* if the -ary semigroup is cancellative.

Recall that for an -ary semigroup , a nonempty subset of is called a subsemigroup of if whenever all . For , we call an -ideal of if whenever . is called an ideal of if and only if it is an -ideal of . See, for example, [7, Definition 1.6].

*Definition 3. *A nonempty subset of an -semiring is called an -ary *subsemiring* of if is a subsemigroup of as well as a subsemigroup of and an *(i-)ideal* of if is a subsemigroup of as well as an *(i-)ideal* of (where ). An -ideal is also called a *right ideal* and an -ideal is also called a *left ideal*. An ideal of is called a *-ideal* if and imply that . An ideal of is called an -ideal if and imply that .

Let be an ideal of . Then the -closure of , denoted by , is defined by . Similarly, the -closure of , denoted by , is defined by for some and some . One can show that is a -ideal and is an -ideal. Furthermore, it is shown that an ideal of is a -ideal if and only if and that is an -ideal if and only if .

*Definition 4. *An equivalence relation on an -semiring is said to be a *congruence relation* or simply a *congruence* of if the following conditions are satisfied:(1) for all , (2) for all .

Let be a proper ideal of an -semiring . Then the congruence on , denoted by , and defined by setting if and only if for some , is called the *Bourne congruence* on defined by the ideal . We denote the Bourne congruence class of an element by and denote the set of all such congruence classes of by . If the Bourne congruence is proper, that is, , then we can define two operations, -ary addition and -ary multiplication on by and for all . Then is an -semiring and is called the *Bourne factor **-semiring*.

Similarly, the congruence on , denoted by , and defined by setting if and only if for some and some , is called the *Iizuka congruence* on defined by the ideal . We denote the Iizuka congruence class of an element by and denote the set of all such congruence classes of by . If the Iizuka congruence is proper, that is, , then we can define two operations, -ary addition and -ary multiplication on by and for all . Then is an -semiring and we call it the *Iizuka factor **-semiring*.

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

*Definition 5. * A commutative -ary semigroup with an identity (operation to be called addition) is called a *right **-ary semimodule* over an -semiring or simply an -ary -semimodule if there exists a mapping (images to be denoted by or briefly by for all and ) satisfying the following conditions: (1) for all and all ; (2) for all and all ; (3) for all , , and ; (4) for all ; (5) whenever , , and for some .

*Definition 6. * A nonempty subset of a right -ary semimodule over an -semiring is called an -ary *subsemimodule* of if (i) and (ii) for all and .

An -ary subsemimodule of is called an -ary -subsemimodule if and imply that . An -ary subsemimodule of is called an -ary -subsemimodule if and imply that .

For example, an -semiring can be regarded as a right -ary -semimodule naturally. Then if is a -ideal (an -ideal) of the -semiring , then is also an -ary -(-)subsemimodule of this right -ary -semimodule .

*Definition 7. *A right -ary -semimodule is said to be *cancellative* if is a cancellative -ary semigroup.

*Definition 8. * An equivalence relation on right -ary -semimodule is said to be a *congruence relation* or simply a *congruence* of if the following conditions are satisfied:(1) for all , (2) for all and all . We say that a congruence of admits the cancellation law (of addition) if (3) and imply .

Let be an -ary subsemimodule of an -ary right semimodule over an -semiring . Then the congruence on , denoted by , and defined by setting
is called the *Bourne congruence* on defined by the -ary subsemimodule . We denote the Bourne congruence class of an element by and denote the set of all such congruence classes of by . Define two operations, -ary addition and -ary scalar multiplication on , by and for all and all . With these two operations, is an -ary right semimodule over and we call it the *Bourne factor **-ary semimodule*.

Similarly, we can define the *Iizuka congruence* and the *Iizuka factor **-ary semimodule* . It is easy to show that is cancellative.

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

#### 3. Primitive -Semirings

*Definition 9. *Let be an -semiring with zero . Then the *zeroid* of , denoted by , is defined as

Clearly, the zero element of belongs to . Furthermore, we have the following.

Lemma 10. * The zeroid of an -semiring is the smallest -ideal of . *

*Proof. *It is easily verified that is an ideal of . To show is an -ideal of , we suppose , where and . For each there exist such that , so we have
that is,

It follows that

Hence we obtain
which shows that , so that is an -ideal of .

At last, suppose that is an arbitrary -ideal of . We aim to show . For this, let . Then there exist such that , so . It follows that since is an -ideal and . Thus .

*Definition 11. *Let be a right -ary -semimodule. The *annihilator* of in , denoted by or , is defined as the subset

Lemma 12. * is an -ideal of . *

*Proof. * It is obvious that is an ideal of . To show that it is an -ideal, suppose , where and . Then for all ,
that is,
which deduces that
since for each . Thus we have

By cancellation law of , . Hence , as required.

*Definition 13. *A right -ary -semimodule is said to be *faithful* if .

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

*Definition 14. * A right -ary -semimodule is said to be *irreducible* if for every arbitrary fixed pair with for some and for any , there exist with such that

*Remark 15. * Since is cancellative, it is easily seen that a right -ary -semimodule is irreducible if and only if for every arbitrary fixed pair with for all and for any , there exist with such that equality (2) holds.

Lemma 16. *Let be an -ideal of an -semiring . If is an irreducible right -ary -semimodule, then is an irreducible right -ary -semimodule. *

*Proof. * Let be an irreducible right -ary -semimodule. Then we can define an -ary action on by for all and for all , and this makes into an irreducible right -ary -semimodule.

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

Lemma 17. *If is a right -ary -semimodule then is a right -ary -semimodule, where is the Bourne factor semiring. Moreover, if is an irreducible right -ary -semimodule, then is also an irreducible right -ary -semimodule. *

*Proof. *Suppose is a right -ary -semimodule. We define an -ary action on as follows: where , for all and for all . We now show that this definition is well-defined. If for each , , then , that is, there exist such that . It follows that , which implies that since . Thus , as required. It is easy to see that the above definition makes into a right -ary -semimodule.

Moreover, if is an irreducible right -ary -semimodule then it is routine to verify that is also an irreducible right -ary -semimodule by (2).

Lemma 18. * Let be a right -ary -semimodule. Then . *

*Proof. * Let , where . Then for any whenever and for some . It follows that where for some . This shows that and so that . Consequently, .

Lemma 19. * Any right -ary -semimodule is a faithful -semimodule. *

*Proof. *Let be a right -ary -semimodule. Then in view of Lemma 17, is an -semimodule. On the one hand, by Lemma 12, is an -ideal of . On the other hand, by Lemma 10, is the smallest -ideal of . Thus . According to Lemma 18, . So , which means that is a faithful -semimodule.

Lemma 20. * If is an -ideal of an -semiring then where is the Bourne factor semiring. *

*Proof. *Suppose . Then we have for some . Thus we have which implies that for some . Hence . This shows that since is an -ideal of . Consequently, . Thus .

*Definition 21. *An -semiring is said to be *primitive* if it has a faithful irreducible cancellative -ary -semimodule. An ideal is said to be *primitive* if the Bourne factor semiring is primitive.

Evidently, an -semiring is primitive if and only if is a primitive ideal of . The following theorem characterizes primitive ideals of an -semiring.

Theorem 22. * An -ideal of -semiring is primitive if and only if for some irreducible right -ary -semimodule . *

*Proof. *Let be an -ideal of such that for some irreducible right -ary -semimodule . Then by Lemmas 17 and 19 is a faithful irreducible -ary -semimodule. This shows that is primitive and hence is a primitive -ideal of .

Conversely, let be a primitive -ideal of . Then is a primitive -semiring. So there exists a faithful irreducible -ary -semimodule . Now by Lemma 16 is an irreducible -ary -semimodule. It remains to show that . Now for all and , whenever for some whenever for some since is a faithful -ary -semimodule , by Lemma 20 . Thus as desired.

#### 4. Jacobson Radical of an -Semiring

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

*Definition 23. *A right -ary -semimodule is said to be *semi-irreducible* if ; that is, for some and some , and does not contain any -ary -subsemimodule other than and .

Lemma 24. *Let be a subset of an -semiring and a right -ary -semimodule with for some . In the case where , we assume further that is a left ideal of . Then the following statements are true:*(1)*If is semi-irreducible and , then if and only if ; *(2)*If is irreducible and , then if and only if for all and all . *

*Proof. *Suppose that is a semi-irreducible right -ary semimodule over an -semiring , and that is a subset of such that for some . In the case where , we further assume that is a left ideal of .

Assume that is semi-irreducible. Let be such that

Set

It is clear that , and it is easy to show that is a subsemimodule of . Let and . Then and . Thus ; that is, . This shows that is a -subsemimodule of . Since , . Since is semi-irreducible, and therefore .

The converse part is obvious.

Assume that is irreducible. Let be such that . Set , for . Since , we have for some and . Since is irreducible, according to Definition 14, there exist with such that

Hence

Since is cancellative and , at least one of the following equalities does not hold:

So we conclude that if for all and all , then .

The converse part follows easily.

Lemma 25. *Let be a right -ary -semimodule and . Then is semi-irreducible if and only if for every nonzero , . *

*Proof. *Assume that is a semi-irreducible right -ary -semimodule and . Let be such that . Then by Lemma 24 . Since is an -ary -subsemimodule of , .

Conversely, suppose that for any nonzero , . Let be an -ary -subsemimodule of . Then there exists such that . So by hypothesis, . Hence for any , there exist such that . Since , we have . Since is an -ary -subsemimodule, implies that . This shows that . Now if then for all . Hence . So we have , a contradiction. Therefore, . Thus is semi-irreducible.

Corollary 26. * If a right -ary -semimodule is irreducible, then it is semi-irreducible and . *

*Proof. * Assume that is an irreducible right -ary -semimodule. Then and, consequently, there exists a nonzero . In view of (2) with and for , we obtain that for any there exist with such that
so that

It follows that . Thus . By Lemma 25, is semi-irreducible.

Furthermore, , which implies that . Since is an -ary -subsemimodule of , as required.

Now we can define the Jacobson radical of an -semiring in an external way.

*Definition 27. *Let be an -semiring and be the set of all irreducible right -ary -semimodules. Then is called the *Jacobson radical* of . If is empty then itself is considered as ; that is, , and in this case, we say that is a *radical **-semiring*. An -semiring is said to be *Jacobson semisimple* or *J-semisimple* if .

By Lemma 12, is an -ideal of . Note that the intersection of any family of -ideals is again an -ideal. Consequently, we obtain the following.

Lemma 28. * is an -ideal of . *

Lemma 29. * If is a right -ary -semimodule then is a right -ary -semimodule, where is the Bourne factor semiring. Moreover, if is an irreducible right -ary -semimodule, then is also an irreducible right -ary -semimodule. *

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

Theorem 30. * If is an -semiring, then the Bourne factor semiring is Jacobson semisimple.*

*Proof. *By and , we denote the set of all irreducible right -ary -semimodules and the set of all irreducible right -ary -semimodules, respectively. Then according to Lemmas 28, 16, and, 29, we obtain that . For any and any , we have , which means that for any , whenever and for some . Thus whenever and for some , so for all . That is, . Hence . We have shown that . By Definition 27, is Jacobson semisimple.

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

Theorem 31. * is the intersection of all primitive -ideals of . *

*Definition 32. * Let be an -ideal of an -semiring for some . Then is said to be *strongly seminilpotent* if there exists a positive integer such that , where , times, . is said to be *strongly nilpotent* if there exists a positive integer such that .

Theorem 33. * If is a strongly semi-nilpotent left ideal of , then . *

*Proof. *Suppose on the contrary that
where is an -semiring and is the set of all irreducible right -ary -semimodules. Then there exists an such that . Thus there exists such that

Since is strongly semi-nilpotent, there exists a positive integer such that . By Lemmas 10 and 12, . It follows that , which implies that
If (29) holds for all positive integers ’s, then in particular it is true for and in this case we have , a contradiction to (28). If (29) does not hold for all , then there exist and positive such that
Thus and there exists such that . It follows that

so we have

By Lemma 24, we obtain , again a contradiction. This completes the proof.

The next result is a direct corollary of Theorem 33.

Corollary 34. *If an -semiring is Jacobson semisimple then does not contain any non-zero strongly semi-nilpotent left ideal and hence does not contain any nontrivial strongly nilpotent left ideal. *

Lemma 35. *If is a (semi-)irreducible right -ary -semimodule and is an arbitrary -subsemimodule (and ), then is (semi-)irreducible, and for any the following statement is true: the equality holds for all if and only if the same equality holds for all . Furthermore, . *

*Proof. *Assume is an irreducible right -ary -semimodule. Then from (2), it follows that is irreducible. If is a semi-irreducible and , then is semi-irreducible by Definition 23 since any subsemimodule of is clearly a subsemimodule of .

Let be such that the equality holds for all . Since is semi-irreducible, for any and any , there exist positive , and such that and

Thus we have the following two equalities:

It follows that

Observing that , since is a submodule, we have and for all by the assumption. Hence by cancellation law, (35) deduces that . The converse implication is clear.

Furthermore, letting for some , we get that the equality holds for all if and only if the same equality holds for all . Thus .

Lemma 36. * Let be an ideal of an -semiring . *(1)*If is an (semi-)irreducible right -ary -semimodule (and ), then is an (semi-)irreducible right -ary -semimodule. *(2)*If is an irreducible right -ary -semimodule, then there exists an irreducible right -ary -semimodule , which can be regarded as an -subsemimodule of . *

*Proof. * Let be an irreducible -semimodule and be such that for some . Without loss of generality, we suppose that . From (2) we deduce that . By Lemma 24, for some . Since is irreducible, by (19) there exist with such that
that is,
which means that is an irreducible -semimodule by (19) again since for all .

Assume that is a semi-irreducible -semimodule and . According to Lemma 24, for any there exist such that . By Lemma 25, , so for any there exist positive integers and such that and
which shows that

Note that for all , , . Thus we obtain . By Lemma 25 again, is a semi-irreducible right -ary -semimodule.

Let be an irreducible right -ary -semimodule, and let . Then and is an -subsemimodule of . Thus by Lemma 35, is irreducible and for any the following conclusion is true: the equality holds for all if and only if the same equality holds for all . If for some and , then for any and any ,