# On Exact Sequence of Semimodules over Semirings

Authors: Jayprakash Ninu Chaudhari; Dipak Ravindra Bonde
Journal: ISRN Algebra
 (2013)

---

## 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, “If0→L→fM→gN→0 is a split exact sequence of R-semimodules and maximal R-semimodule homomorphisms with Ker(g) is a strong subsemimodule of M, then M≅L⊕N”. Also, some results of commutative diagram of R-semimodules and maximal R-homomorphisms having exact rows are obtained.


---

## Body

## 1. Introduction

The concept of semiring was introduced by Vandiver [1] in 1934. A semiring is a nonempty set R together with two associative binary operations, addition and multiplication, which is called a semiring if (i) addition is a commutative operation, (ii) there exists 0∈R such that x+0=x=0+x, x0=0=0x for each x∈R, and (iii) multiplication distributes over addition both from left and right. Denote the sets of all nonnegative and residue classes of integers modulo n, respectively, by ℤ0+ and ℤn. The set ℤ0+ is a semiring under usual addition and multiplication of nonnegative integers, but it is not a ring. Concepts of commutative semiring, semiring with identity 1, can be defined on the similar lines as in rings. All semirings in this paper are assumed to be commutative with identity 1≠0.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 R-semimodules and maximal R-homomorphisms having exact rows are obtained.The following definitions and results will be used to prove our results.A leftR-semimodule is a commutative monoid (M,+) with additive identity 0M for which we have a function R×M→M, defined by  (r,x)↦rx and called scalar multiplication, which satisfies the following conditions for all elements r and r′ of R and all elements x and y of M: (1)
(rr′)x=r(r′x);(2)
r(x+y)=rx+ry;(3)
(r+r′)x=rx+r′x;(4)
1Rx=x;(5)
r0M=0M=0Rx.Throughout this paper, by anR-semimodule we mean a left semimodule over a semiring R. Every semiring R is a (ℤ0+,+,·)-semimodule (see [5, page 151]). An element a of a monoid (M,*) is called idempotent if a*a=a. If (M,+) is an idempotent commutative monoid, then M is a (ℤ0+,+,·)-semimodule with scalar multiplication defined by rm=0 if r=0 and rm=m if r>0 for all r∈ℤ0+ and m∈M (see [5, page 151]). A nonempty subset N of an R-semimodule M is called a subsemimodule of M if N is closed under addition and closed under scalar multiplication. A subsemimodule N of an R-semimodule M is called (1) a subtractive subsemimodule (= k-subsemimodule) if x, x+y∈N, y∈M, then y∈N; (2) a strong subsemimodule if for any x∈N there exists y∈N, such that x+y=0.Chaudhari and Bonde [6] introduced the notion of a partitioning subsemimodule and developed the quotient structure of semimodules over semirings. A subsemimodule N of an R-semimodule M is called partitioning subsemimodule (= Q-subsemimodule) if there exists a subset Q of M such that M = ∪{q+N:q∈Q}, and if q1, q2∈Q, then (q1+N)∩(q2+N)≠∅⇔q1=q2. If N is a subsemimodule of a (ℤ0+,+,·)-semimodule (ℤ0+, +) and N is generated by n, then N is a partitioning subsemimodule with Q={0,1,…,n-1} [7, Example 2.2(2)]. If N is a partitioning subsemimodule of an R-semimodule M, then M/N(Q)={q+N:q∈Q} forms an R-semimodule under the following addition “⊕” and scalar multiplication “⊙”, (q1+N)⊕(q2+N)=q3+N where q3∈Q is unique such that q1+q2+N⊆q3 + N, and r⊙(q1+N)=q4+N where q4∈Q is unique such that rq1+N⊆q4 + N. This R-semimodule M/N(Q) is called a quotient semimodule of M by N and denoted by M/N(Q). By Lemma 2.3 ([6]), there exists a unique q0∈Q such that q0+N=N. This q0+N is the zero element of M/N(Q). An R-semimodule M is said to be a direct sum of subsemimodules N1,N2,…,Nk of M, if each x∈M can be uniquely written as x=x1+x2+⋯+xk where xi∈Ni, 1≤i≤k. It is denoted by M=N1⊕N2⊕⋯⊕Nk (see [5, page 184]). If R is a semiring and M, M′ are R-semimodules, then a function f:M→M′ is called an R-semimodule homomorphism if f(x+y)=f(x)+f(y) for all x,y∈M and f(rx)=rf(x) for all x∈M and r∈R. An equivalence relation ρ on an R-semimodule M is said to be an R-congruence relation if aρb implies that (a+c)ρ(b+c) for all c∈M and (ra)ρ(rb) for all r∈R. If f:M→M′ is an R-semimodule homomorphism, then f induces an R-congruence relation ≡f on M given by m≡fm′⇔f(m)=f(m′) where m,m′∈M. If N is a subsemimodule of an R-semimodule M, then N induces an R-congruence relation ≡N on M given by m≡Nm′⇔ there exist n,n′∈N such that m+n=m′+n′ where m,m′∈M. If f:M→M′ is an R-semimodule homomorphism, then clearly m≡Ker(f)m′ implies that m≡fm′ where m,m′∈M and Ker(f)={x∈M:f(x)=0′}.Definition 1.
Letf:M→M′ be an R-semimodule homomorphism. Then, f is called(1)
Steady (ork-uniform [2] or k-regular [3]) if ≡f coincides with ≡Ker(f);(2)
maximal [6] if for each a∈f(M), there exists a unique qa∈f-1({a}) such that x+Ker(f)⊆qa+Ker(f) for all x∈f-1({a});(3)
one-one iff(x)=f(y) implies x=y where x,y∈M;(4)
isomorphism iff is one-one and onto;(5)
i-regular [3] (or i-uniform [2]) if Im(f) is a subtractive subsemimodule of M′.Clearly, identityR-semimodule homomorphism from M onto itself is a maximal R-semimodule homomorphism.Lemma 2 (see [8, Lemma 2.2]).
Iff:M→M′ is a maximal R-semimodule homomorphism, then f is steady.Theorem 3 (see [8, Theorem 2.1]).
Letf:M→M′ be an R-semimodule homomorphism. Then, the following statements are equivalent: (1)
f is one-one;(2)
Ker(f)={0} and f is maximal;(3)
Ker(f)={0} and f is steady.Theorem 4 (see [8, Theorem 3.3]).
LetN1,N2 be subsemimodules of an R-semimodule M. Then, M=N1⊕N2 if and only if N1,N2 are partitioning subsemimodules, M=N1+N2, and N1∩N2={0}.


## 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 sequenceA→fB→gC of R-semimodules and R-semimodule homomorphisms is said to be exact if (1)
Im(f)=Ker(g) and g is k-uniform [2];(2)
If¯=Kg where If¯={(c,d)∈B×B:c+f(a)+x=d+f(b)+x  for  some  a,b∈A,x∈B} and Kg={(a,b)∈B×B:f(a)+x=f(b)+x  for  some  x∈B} [4];(3)
f(A)=Ker(g) and proper exact if Ker(g) = subtractive closure of f(A) [3].Now, we introduce the notion of exact sequence of semimodules over semirings using maximal homomorphisms, and hence, some results of commutative diagram ofR-semimodules and maximal R-homomorphisms having exact rows are obtained.Definition 6.
A sequenceM0→f1M1→f2M2→f3⋯→fnMn of R-semimodules and maximal R-semimodule homomorphisms is said to be exact if Im(fi)=Ker(fi+1) for all 1≤i≤n-1.As every ring,R-module, and R-module homomorphism is, respectively, semiring, R-semimodule, and maximal R-semimodule homomorphism, we have every exact sequence of R-modules and R-homomorphisms is an exact sequence of R-semimodules and maximal R-semimodule homomorphisms.IfA,B, and C are R-semimodules, then by Theorem 3: (1)
the sequence0→A→ψB is exact if and only if ψ is one-one;(2)
the sequenceB→ϕC→0 is exact if and only if ϕ is maximal and onto.Now, we have the following proposition.Proposition 7.
Letψ:A→B, ϕ:B→C be R-semimodule homomorphisms. Then, the sequence 0→A→ψB→ϕC→0 is exact if and only if ψ is one-one, ϕ is maximal and onto, and Im(ψ)=Ker(ϕ).Definition 8.
An exact sequence0→A→ψB→ϕC→0 of R-semimodules and maximal R-semimodule homomorphisms will be called as a short exact sequence.Definition 9.
A short exact sequence,0→L→fM→gN→0 of R-semimodules and maximal R-semimodule homomorphisms, is said to be split exact sequence if there exists a splitting map g′:N→M such that g′(N) is partitioning subsemimodule of M and gg′=IdN.Example 10.
LetA and B be any R-semimodules. (1)
Ifϕ:A→B is any maximal R-semimodule homomorphism, then 0→Ker(ϕ)→iA→ϕIm(ϕ)→0 is an exact sequence where i is an identity R-semimodule homomorphism.(2)
Consider (ℤ0+,+,·)-semimodules L=(ℤ0+,+), M=(ℤ0+∪{∞},+), and N=({0,1,∞},max), maximal R-semimodule homomorphism g:M→N defined by
(1)g(m)={0ifm≠∞,∞ifm=∞,and an identityR-semimodule homomorphism i:L→M. Then, the sequence 0→L→iM→gN→0 is not an exact sequence because g is not onto.(3)
Consider0→A→iA⊕B→πB→0, a sequence of R-semimodules where i is an identity R-semimodule homomorphism and π is defined by π(a,b)=b, for all (a,b)∈A⊕B is onto maximal R-semimodule homomorphism with Ker(π)=A. This sequence is a split exact sequence with inclusion map t:B→A⊕B as a splitting map, since, by Theorem 4, t(B)=B is a partitioning subsemimodule of A⊕B.(4)
IfN is a partitioning subsemimodule of an R-semimodule A, then f:A→A/N(Q) defined by f(a)=q+N where q∈Q is a unique element such that a+N⊆q+N is onto maximal R-semimodule homomorphism with Ker(f)=N. Now, the sequence 0→N→iA→fA/N(Q)→0 is an exact sequence where i is an identity R-semimodule homomorphism.(5)
LetN=(2ℤ0+,+) be a Q={0,1}-subsemimodule of an R=(ℤ0+,+,·)-semimodule A=(ℤ0+,+). Then, by Example 10(4), the sequence 0→N→iA→fA/N(Q)→0 is an exact sequence but it is not a split exact sequence because the only R-semimodule homomorphism from A/N(Q) to A is a zero R-semimodule homomorphism.Theorem 11.
If0→L→fM→gN→0 is a split exact sequence of R-semimodules and maximal R-semimodule homomorphisms with Ker(g) is a strong subsemimodule of M, then M≅L⊕N.Proof.
Let0→L→fM→gN→0 be a split exact sequence of R-semimodules, and let maximal R-semimodule homomorphisms with a splitting map g′ and Ker(g) be a strong subsemimodule of M. Claim that M=f(L)⊕g′(N)=Ker(g)⊕g′(N). Since g is maximal, Ker(g) is partitioning subsemimodule of M (see [6, Lemma 2.8]). Also, g′(N) is a partitioning subsemimodule of M, since given sequence is a split exact sequence. Hence, by using Theorem 4, it is enough to prove that M=Ker(g)+g′(N) and Ker(g)∩g′(N)={0}. Let m∈M and g(m)=n∈N. Hence, g(m)=n=gg′(n)=g(g′(n)). By Lemma 2, g is steady. Hence, m+k1=g′(n)+k2 for some k1,k2∈Ker(g). Since Ker(g) is a strong subsemimodule of M, there exists k∈Ker(g) such that k1+k=0. Thus, we get m=m+k1+k=g′(n)+k2+k∈Ker(g)+g′(N). So, M=Ker(g)+g′(N)=f(L)+g′(N). Now, f(l)=g′(n)∈f(L)∩g′(N)⇒0=gf(l)=gg′(n)=n⇒g′(n)=0. Hence, f(L)∩g′(N)={0}. So, M=f(L)⊕g′(N). Now, define θ:L⊕N→f(L)⊕g′(N) by θ(l+n)=f(l)+g′(n). Clearly, θ is an onto R-semimodule homomorphism. Since f is one-one and gg′=IdN, θ is one-one. Thus, M≅L⊕N.Corollary 12.
Let0→L→fM→gN→0 be a split exact sequence of R-semimodules, and let maximal R-semimodule homomorphisms with Ker(g) be a strong subsemimodule of M. Then, there exists a maximal R-semimodule homomorphism h:M→L such that hf=IdL.Proof.
By Theorem11, M=Im(f)⊕g′(N). Clearly, π:M→Im(f) defined by π(x+y)=x for all x∈Im(f) and y∈g′(N) is a maximal R-semimodule homomorphism. Let h=f-1π. Then, h is maximal, since f is maximal and hf=IdL.Example 13.
ConsiderL=({0¯,2¯,4¯},+6), M=(ℤ6,+6), N=({0¯,3¯},+6) the R=(ℤ0+,+,·)-semimodules, and consider thatg:M→N defined by g(m¯)=n¯ where n≡3m(mod6) is an R-semimodule homomorphism, f:L→M is an identity R-semimodule homomorphism. Here, g is maximal onto R-semimodule homomorphism. Then, the sequence 0→L→fM→gN→0 is split exact sequence and Ker(g) is a strong subsemimodule of M, and hence, by Theorem 11, M≅L⊕N.The following example shows that the condition “Ker(g) is a strong subsemimodule of M” is essential.Example 14.
LetL=(ℤ0+,+),M=(ℤ0+∪{∞},+), and N=({0,∞},+) be R=(ℤ0+,+,·)-semimodules, and let  0→L→fM→gN→0 be a sequence of R-semimodules and maximal R-semimodule homomorphisms where f is an inclusion map and g:M→N is defined as
(2)g(m)={0ifm≠∞,∞ifm=∞.
Then, clearly,0→L→fM→gN→0 is an exact sequence. Now take g′:N→M an identity R-semimodule homomorphism. Then, gg′=IdN. Here, M=L+N and M≠L⊕N because ∞=a+∞ for all a∈M. Hence M is not isomorphic to L⊕N. Clearly, Ker(g) is not a strong subsemimodule of M.Now, we extend and prove two theorems of commutative diagram ofR-modules and R-homomorphisms having exact rows for commutative diagram of R-semimodules and maximal R-homomorphisms having exact rows.Theorem 15.
Suppose that the following diagram ofR-semimodules and maximal R-semimodule homomorphisms is commutative and has exact rows:
(3)(1)
Ifβ is one-one and α, g are onto, then γ is one-one.(2)
Ifβ is onto and f′, γ are one-one, then α is onto.Proof.
(
1
)Let c∈Ker(γ). Since g is onto, there exists b∈B such that g(b)=c. Now, 0=γ(c)=γg(b)=g′β(b)⇒β(b)∈Ker(g′)=Im(f′)⇒β(b)=f′(a′) for some a′∈A′. Since α is an onto, a′=α(a) for some a∈A. Now, β(b)=f′(a′)=f′α(a)=βf(a). Since β is one-one, b=f(a)∈Im(f)=Ker(g). So c=g(b)=0. Thus, Ker(γ)={0}. Now, γ is a maximal R-semimodule homomorphism, and hence, by Theorem 3, γ is one-one.
(
2
)
Let a′∈A′. Since β is onto, there exists b∈B such that β(b)=f′(a′)∈Im(f′)=Ker(g′). Now, g′β=γg⇒γg(b)=g′β(b)=0⇒g(b)∈Ker(γ)={0} (since γ is one-one) ⇒b∈Ker(g)=Im(f)⇒b=f(a) for some a∈A. Now, f′(a′)=β(b)=βf(a)=f′α(a). Since f′ is one-one, a′=α(a). Hence, α is onto.Theorem 16.
Suppose that the following diagram ofR-semimodules and maximal R-semimodule homomorphisms is commutative and has exact rows and middle column is exact:
(4)
Then, the last column is exact, andg′(B′) is subtractive if and only if the first column is exact.Proof.
Suppose that the last column is exact andg′(B′) is subtractive. We claim that Ker(f′)={0},Im(f′)=Ker(g′), and g′ is onto.
(
i
)Let a′∈Ker(f′). Now, fα′(a′)=β′f′(a′)=β′(0)=0 implies that α′(a′)∈Ker(f)={0}. Since α′ is one-one, a′=0. Hence, Ker(f′)={0}.
(
ii
)Let b′∈Ker(g′). Now, gβ′(b′)=γ′g′(b′)=γ′(0)=0⇒β′(b′)∈Ker(g)=Im(f). So, β′(b′)=f(a) for some a∈A. As Im(β′)=Ker(β) and βf=f′′α, we get 0=ββ′(b′)=βf(a)=f′′α(a). Hence, α(a)∈Ker(f′′)={0} implies that a∈Ker(α)=Im(α′). So a=α′(a′) for some a′∈A′. Now, β′(b′)=f(a)=fα′(a′)=β′f′(a′). Since β′ is one-one, b′=f′(a′)∈Im(f′). Hence, Ker(g′)⊆Im(f′). Now, let x=f′(a′)∈Im(f′) for some a′∈A′. Therefore, β′(x)=β′f′(a′)=fα′(a′) implies that β′(x)∈Im(f)=Ker(g). Now, γ′g′(x)=gβ′(x)=0. Since γ′ is one-one, g′(x)=0. Now x∈Ker(g′). Thus, Im(f′)⊆Ker(g′). Hence, we get the required equality.
(
iii
)Let c′∈C′. Therefore, γ′(c′)∈C and γ′(c′)=g(b) for some b∈B, since g is onto. Now, g′′β(b)=γg(b)=γγ′(c′)=0 implies that  β(b)∈Ker(g′′)=Im(f′′) implies β(b)=f′′(a′′) for some a′′∈A′′. As α is onto, we get β(b)=f′′(a′′)=f′′α(a)=βf(a) for some a∈A. Since β is maximal and hence steady, so there exist k1,k2∈Ker(β) such that b+k1=f(a)+k2. As Ker(β)=Im(β′), we have b+β′(b1′)=f(a)+β′(b2′)⇒g(b)+gβ′(b1′)=gf(a)+gβ′(b2′)⇒g(b)+γ′g′(b1′)=0+γ′g′(b2′)⇒γ′(c′+g′(b1′))=γ′g′(b2′)⇒c′+g′(b1′)=g′(b2′), since γ′ is one-one. Now, c′∈g′(B′), as g′(B′) is subtractive. Hence, g′ is onto. So, 0→A′→f′B′→g′C′→0 is an exact sequence.
Conversely, assume that the first column is exact, then, by a similar argument as previously mentioned we can show thatKer(f′′)={0},Im(f′′)=Ker(g′′). Now, let c′′∈C′′. Since γ and g are onto, there exists b∈B such that c′′=γg(b)=g′′β(b)∈Im(g′′). So, g′′ is onto. Hence, the last column is exact. Also, g′ is onto which implies that g′(B′)=C′ which is a subtractive subsemimodule.Now, we extend and prove the well-known “5-lemma [9]” of R-modules and R-homomorphisms for R-semimodules and maximal R-semimodule homomorphisms.Lemma 17 (the 5-lemma).
Given commutative diagram ofR-semimodules and maximal R-semimodule homomorphisms with exact rows:
(5)
Then,(i)
iff1 is onto and f2,f4 are one-one, then f3 is one-one;(ii)
iff5 is one-one, f2,f4 are onto, and Im(f3) is a subtractive subsemimodule of N3, then f3 is onto;(iii)
iff1,f2,f4,f5 are isomorphisms and Im(f3) is a subtractive subsemimodule of N3, then so is f3.Proof.
(i) Letm3∈Ker(f3). Now, f4g3(m3)=h3f3(m3)=0⇒g3(m3)∈Ker(f4)={0}⇒m3∈Ker(g3)=Im(g2)⇒m3=g2(m2) for some m2∈M2. Now, 0=f3(m3)=f3g2(m2)=h2f2(m2)⇒f2(m2)∈Ker(h2)=Im(h1)⇒f2(m2)=h1(n1) for some n1∈N1. Since f1 is onto, n1=f1(m1) for some m1∈M1. Thus, f2(m2)=h1f1(m1)=f2g1(m1). So, m2=g1(m1), since f2 is one-one. Now, m3=g2(m2)=g2g1(m1)=0. So, Ker(f3)={0}. Since f3 is maximal, f3 is one-one.
(ii) Letn3∈N3. Since f4 is onto, h3(n3)=f4(m4) for some m4∈M4. Now, 0=h4h3(n3)=h4f4(m4)=f5g4(m4)⇒g4(m4)=0 since f5 is one-one. So, m4∈Ker(g4)=Im(g3)⇒m4=g3(m3) for some m3∈M3. Now, h3(n3)=f4(m4)=f4g3(m3)=h3f3(m3) and h3 is maximal and hence steady which implies that there exist k1=h2(n2′),k2=h2(n2′′)∈Ker(h3)=Im(h2) for some n2′,n2′′∈N2 such that n3+h2(n2′)=f3(m3)+h2(n2′′). Now n2′=f2(m2′) and n2′′=f2(m2′′) for some m2′,m2′′∈M2, since f2 is onto. Now, n3+h2(n2′)=f3(m3)+h2(n2′′)⇒n3+h2f2(m2′)=f3(m3)+h2f2(m2′′)⇒n3+f3g2(m2′)=f3(m3)+f3g2(m2′′)∈Im(f3). Since Im(f3) is subtractive subsemimodule of N3, n3∈Im(f3). Hence, f3 is onto.
(iii) Follows from (i) and (ii).



---

*Source: 156485-2013-07-31.xml*