Abstract

We introduce the notion of exact sequence of semimodules over semirings using maximal homomorphisms and generalize some results of module theory to semimodules over semirings. Indeed, we prove, “If is a split exact sequence of -semimodules and maximal -semimodule homomorphisms with is a strong subsemimodule of , then ”. Also, some results of commutative diagram of -semimodules and maximal -homomorphisms having exact rows are obtained.

1. Introduction

The concept of semiring was introduced by Vandiver [1] in 1934. A semiring is a nonempty set together with two associative binary operations, addition and multiplication, which is called a semiring if (i) addition is a commutative operation, (ii) there exists such that , for each , and (iii) multiplication distributes over addition both from left and right. Denote the sets of all nonnegative and residue classes of integers modulo , respectively, by and . The set is a semiring under usual addition and multiplication of nonnegative integers, but it is not a ring. Concepts of commutative semiring, semiring with identity , can be defined on the similar lines as in rings. All semirings in this paper are assumed to be commutative with identity .

Generalizing the notion of exact sequence of modules over rings, Abuhlail [2], Al-Thani [3], and Bhambri and Dubey [4] introduced different notions of exact sequence of semimodules over semirings. In this paper, we introduce the notion of exact sequence of semimodules over semirings using maximal homomorphisms, and hence, some results of commutative diagram of -semimodules and maximal -homomorphisms having exact rows are obtained.

The following definitions and results will be used to prove our results.

A left -semimodule is a commutative monoid with additive identity for which we have a function , defined by and called scalar multiplication, which satisfies the following conditions for all elements and of and all elements and of : (1); (2); (3); (4); (5).

Throughout this paper, by an -semimodule we mean a left semimodule over a semiring . Every semiring is a -semimodule (see [5, page 151]). An element of a monoid is called idempotent if . If is an idempotent commutative monoid, then is a -semimodule with scalar multiplication defined by if and if for all and (see [5, page 151]). A nonempty subset of an -semimodule is called a subsemimodule of if is closed under addition and closed under scalar multiplication. A subsemimodule of an -semimodule is called (1) a subtractive subsemimodule (= -subsemimodule) if , , , then ; (2) a strong subsemimodule if for any there exists , such that .

Chaudhari and Bonde [6] introduced the notion of a partitioning subsemimodule and developed the quotient structure of semimodules over semirings. A subsemimodule of an -semimodule is called partitioning subsemimodule (= -subsemimodule) if there exists a subset of such that = , and if , , then . If is a subsemimodule of a -semimodule (, +) and is generated by , then is a partitioning subsemimodule with [7, Example ]. If is a partitioning subsemimodule of an -semimodule , then forms an -semimodule under the following addition “” and scalar multiplication “”, where is unique such that + , and where is unique such that + . This -semimodule is called a quotient semimodule of by and denoted by . By Lemma 2.3 ([6]), there exists a unique such that . This is the zero element of . An -semimodule is said to be a direct sum of subsemimodules of , if each can be uniquely written as where , . It is denoted by (see [5, page 184]). If is a semiring and , are -semimodules, then a function is called an -semimodule homomorphism if for all and for all and . An equivalence relation on an -semimodule is said to be an -congruence relation if implies that for all and for all . If is an -semimodule homomorphism, then induces an -congruence relation on given by where . If is a subsemimodule of an -semimodule , then induces an -congruence relation on given by there exist such that where . If is an -semimodule homomorphism, then clearly implies that where and .

Definition 1. Let be an -semimodule homomorphism. Then, is called(1)Steady (or -uniform [2] or -regular [3]) if coincides with ; (2)maximal [6] if for each , there exists a unique such that for all ; (3)one-one if implies where ; (4)isomorphism if is one-one and onto; (5)-regular [3] (or -uniform [2]) if is a subtractive subsemimodule of .

Clearly, identity -semimodule homomorphism from onto itself is a maximal -semimodule homomorphism.

Lemma 2 (see [8, Lemma 2.2]). If is a maximal -semimodule homomorphism, then is steady.

Theorem 3 (see [8, Theorem 2.1]). Let be an -semimodule homomorphism. Then, the following statements are equivalent: (1) is one-one; (2) and is maximal; (3) and is steady.

Theorem 4 (see [8, Theorem 3.3]). Let be subsemimodules of an -semimodule . Then, if and only if , are partitioning subsemimodules, , and .

2. Exact Sequence of Semimodules over Semirings

Generalizing the notion of exact sequence of modules over rings, Abuhlail [2], Al-Thani [3], and Bhambri and Dubey [4] introduced different notions of exact sequence of semimodules over semirings as follows.

Definition 5. A sequence of -semimodules and -semimodule homomorphisms is said to be exact if (1) and is -uniform [2]; (2) where for some and for some [4]; (3) and proper exact if = subtractive closure of [3].

Now, we introduce the notion of exact sequence of semimodules over semirings using maximal homomorphisms, and hence, some results of commutative diagram of -semimodules and maximal -homomorphisms having exact rows are obtained.

Definition 6. A sequence of -semimodules and maximal -semimodule homomorphisms is said to be exact if for all .

As every ring, -module, and -module homomorphism is, respectively, semiring, -semimodule, and maximal -semimodule homomorphism, we have every exact sequence of -modules and -homomorphisms is an exact sequence of -semimodules and maximal -semimodule homomorphisms.

If , and are -semimodules, then by Theorem 3: (1)the sequence is exact if and only if is one-one;(2)the sequence is exact if and only if is maximal and onto.

Now, we have the following proposition.

Proposition 7. Let , be -semimodule homomorphisms. Then, the sequence is exact if and only if is one-one, is maximal and onto, and .

Definition 8. An exact sequence of -semimodules and maximal -semimodule homomorphisms will be called as a short exact sequence.

Definition 9. A short exact sequence, of -semimodules and maximal -semimodule homomorphisms, is said to be split exact sequence if there exists a splitting map such that is partitioning subsemimodule of and .

Example 10. Let and be any -semimodules. (1) If is any maximal -semimodule homomorphism, then is an exact sequence where is an identity -semimodule homomorphism. (2) Consider ()-semimodules , , and , maximal -semimodule homomorphism defined by  and an identity -semimodule homomorphism . Then, the sequence is not an exact sequence because is not onto.(3) Consider , a sequence of -semimodules where is an identity -semimodule homomorphism and is defined by , for all is onto maximal -semimodule homomorphism with . This sequence is a split exact sequence with inclusion map as a splitting map, since, by Theorem 4, is a partitioning subsemimodule of . (4) If is a partitioning subsemimodule of an -semimodule , then defined by where is a unique element such that is onto maximal -semimodule homomorphism with . Now, the sequence is an exact sequence where is an identity -semimodule homomorphism. (5) Let be a -subsemimodule of an -semimodule . Then, by Example 10(4), the sequence is an exact sequence but it is not a split exact sequence because the only -semimodule homomorphism from to is a zero -semimodule homomorphism.

Theorem 11. If is a split exact sequence of -semimodules and maximal -semimodule homomorphisms with is a strong subsemimodule of , then .

Proof. Let be a split exact sequence of -semimodules, and let maximal -semimodule homomorphisms with a splitting map and be a strong subsemimodule of . Claim that . Since is maximal, is partitioning subsemimodule of (see [6, Lemma 2.8]). Also, is a partitioning subsemimodule of , since given sequence is a split exact sequence. Hence, by using Theorem 4, it is enough to prove that and . Let and . Hence, . By Lemma 2, is steady. Hence, for some . Since is a strong subsemimodule of , there exists such that . Thus, we get . So, . Now, . Hence, . So, . Now, define by . Clearly, is an onto -semimodule homomorphism. Since is one-one and , is one-one. Thus, .

Corollary 12. Let be a split exact sequence of -semimodules, and let maximal -semimodule homomorphisms with be a strong subsemimodule of . Then, there exists a maximal -semimodule homomorphism such that .

Proof. By Theorem 11, . Clearly, defined by for all and is a maximal -semimodule homomorphism. Let . Then, is maximal, since is maximal and .

Example 13. Consider , , the -semimodules, and consider that defined by where is an -semimodule homomorphism, is an identity -semimodule homomorphism. Here, is maximal onto -semimodule homomorphism. Then, the sequence is split exact sequence and is a strong subsemimodule of , and hence, by Theorem 11, .

The following example shows that the condition “ is a strong subsemimodule of ” is essential.

Example 14. Let ,, and be -semimodules, and let be a sequence of -semimodules and maximal -semimodule homomorphisms where is an inclusion map and is defined as
Then, clearly, is an exact sequence. Now take an identity -semimodule homomorphism. Then, . Here, and because for all . Hence is not isomorphic to . Clearly, is not a strong subsemimodule of .

Now, we extend and prove two theorems of commutative diagram of -modules and -homomorphisms having exact rows for commutative diagram of -semimodules and maximal -homomorphisms having exact rows.

Theorem 15. Suppose that the following diagram of -semimodules and maximal -semimodule homomorphisms is commutative and has exact rows: (3)(1)If is one-one and , are onto, then is one-one. (2)If is onto and , are one-one, then is onto.

Proof. Let . Since is onto, there exists such that . Now, for some . Since is an onto, for some . Now, . Since is one-one, . So . Thus, . Now, is a maximal -semimodule homomorphism, and hence, by Theorem 3, is one-one.
Let . Since is onto, there exists such that . Now, (since is one-one) for some . Now, . Since is one-one, . Hence, is onto.

Theorem 16. Suppose that the following diagram of -semimodules and maximal -semimodule homomorphisms is commutative and has exact rows and middle column is exact: (4)
Then, the last column is exact, and is subtractive if and only if the first column is exact.

Proof. Suppose that the last column is exact and is subtractive. We claim that ,, and is onto.
Let . Now, implies that . Since is one-one, . Hence, .
Let . Now, . So, for some . As and , we get . Hence, implies that . So for some . Now, . Since is one-one, . Hence, . Now, let for some . Therefore, implies that . Now, . Since is one-one, . Now . Thus, . Hence, we get the required equality.
Let . Therefore, and for some , since is onto. Now, implies that implies for some . As is onto, we get for some . Since is maximal and hence steady, so there exist such that . As , we have =, since is one-one. Now, , as is subtractive. Hence, is onto. So, is an exact sequence.
Conversely, assume that the first column is exact, then, by a similar argument as previously mentioned we can show that ,. Now, let . Since and are onto, there exists such that . So, is onto. Hence, the last column is exact. Also, is onto which implies that which is a subtractive subsemimodule.

Now, we extend and prove the well-known “5-lemma [9]” of -modules and -homomorphisms for -semimodules and maximal -semimodule homomorphisms.

Lemma 17 (the 5-lemma). Given commutative diagram of -semimodules and maximal -semimodule homomorphisms with exact rows: (5)
Then,(i)if is onto and , are one-one, then is one-one; (ii)if is one-one, , are onto, and is a subtractive subsemimodule of , then is onto; (iii)if ,,, are isomorphisms and is a subtractive subsemimodule of , then so is .

Proof. (i) Let . Now, ⇒ for some . Now, = for some . Since is onto, for some . Thus, . So, , since is one-one. Now, . So, . Since is maximal, is one-one.
(ii) Let . Since is onto, for some . Now, since is one-one. So, for some . Now, and is maximal and hence steady which implies that there exist , for some such that . Now and for some , since is onto. Now, ⇒. Since is subtractive subsemimodule of , . Hence, is onto.
(iii) Follows from (i) and (ii).