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Algebra
Volume 2013 (2013), Article ID 272104, 9 pages
http://dx.doi.org/10.1155/2013/272104
Research Article

On the Jacobson Radical of an -Semiring

School of Mathematics and Information Science, Yantai University, Yantai City 264005, China

Received 28 March 2013; Accepted 9 July 2013

Academic Editor: Yiqiang Zhou

Copyright © 2013 Yongwen Zhu. 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 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 [36], 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, [814]), specially on the radical of a semiring; see [1518]. 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 [2224]. 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 [2224], 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 [1518], 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 , which implies that by Lemma 24 since is an irreducible right -ary -semimodule. Thus we can define an operation on into by setting where and . Thus with the addition and the above operation becomes a right -ary -semimodule which, as a right -ary -semimodule, is isomorphic to the right -ary -semimodule . It is clear that is an irreducible right -ary -semimodule.

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

Theorem 37. If is an ideal of an -semiring , then .

Proof. Let be an -semiring and let be the set of all irreducible right -ary -semimodules. Then by Definition 27  . If is an ideal of an -semiring , then , where is the set of all irreducible right -ary -semimodules.
For any , according to Lemma 36, we have . It is evident that . This shows that .
For any , according to Lemma 36, we have that and that