/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 9, No. 3, 2016, 250-265 ISSN 1307-5543 – www.ejpam.com On "Essential" Subsemimodules and Weakly Co-Hopfian Semimodules El Hadji Demba Wade Diop∗, Djiby Sow Department de Mathematiques et Informatique, Faculté des Sciences et Techniques, Université Cheikh Anta Diop, BP 5005 Dakar Fann, Sénégal Abstract. Two different notions of "essential" subsemimodules were introduced in the theory of semi- modules over a semiring with identity, in order to generalize the same notion of "essential" submodules in the theory of modules over a ring with identity. In this paper, we introduce a new class of essential subsemimodules called weakly essential subsemi- modules. We prove that this new class contains the others two kind of classes of "essential" subsemi- modules. Futhermore, we studie the properties of weakly essential subsemimodules. For applications we introduce and investigate the co-hopfian semimodules with this new definition of semi-essential. 2010 Mathematics Subject Classifications: 16Y60 Key Words and Phrases: Semiring, subsemimodule, R-congruence relation, essential, cancellative, subtractive, direct sum, cohopfian semimodules 1. Introduction The theory of semimodules over semirings with identity (see Golan [6], Abuhlail [1], Taka- hashi [9–11]) can be regarded as a generalization of the theory of modules over rings with identity (see Anderson-Fuller[2], Lam [8], Wisbauer [15] ). Many results for semirings and semimodules also hold for rings and modules, but not conversely [5, 12–14]. The concept of "essential submodule" in an R-module M [2], introduced by Johnson, Eck- man and Schpof [4, 7], plays an important role in the context of commutative and noncom- mutative algebras. In module theory, for a ring R, a submodule N of a module M is said to be essential (denoted by N Ã M) if K ∩N = 0=⇒ K = 0 for all submodule K of M . A monomorphism f : M −→ M ′ is said to be essential if f (M)Ã M ′. It is known that an R-monomorphism f : M −→ M ′ of left R-modules is essential iff for any R-homomorphism g : M ′ −→ M”, g ◦ f is a monomorphism implies that g is a monomorphism. A submodule N of a module M is essential iff the injective map i : N −→ M : x 7→ x is essential. ∗Corresponding author. Email addresses: elhjoop@gmail.com (E. Diop), sowdjibab@yahoo.fr (D. Sow) http://www.ejpam.com 250 c© 2016 EJPAM All rights reserved. E. Diop, D. Sow / Eur. J. Pure Appl. Math, 9 (2016), 250-265 251 The previous characterisations of "essential" are not equivalents in semimodules theory. Hence this notion of "essential" was generalized in semimodules theory in two different ways. In Golan book’s [6], it was proposed the following definitions. An R-monomorphism f : M −→ M ′ of left R-semimodules is essential if for any R-homomorphism g : M ′ −→ M ′′, g ◦ f is a monomorphism implies that g is a monomorphism. A subsemimodule N of a left R-semimodule M is essential (or large) in M if the inclusion map iN : N −→ M is an essential R-monomorphism. Note that f : M −→ M ′ is an essential R-homomorphism if and only if f (M) is a large subsemimodule of M ′. Another way for defining the notion of "essential" is proposed in [5] as follows. A left R-subsemimodule N of M is said to be semi-essential in M , written as N Ãs M , if for every R- subsemimodule K of M : N ∩K = 0⇒ K = 0. A monomorphism (respectively: semimonomor- phism) f : M −→ M ′ of left R-semimodules is said to be semi-essential if: f (M)Ãs M ′. These two differents notions of "essential subsemimodules" in the theory of semimodules (see [5]) are the same in the theory of modules. Also, it was proved [5] that the class of essential subsemimodules is not contained and doesn’t contain the class of semi-essential subsemimodules. Furthermore, the intersection of these two classes is not empty. In this paper, we investigate a new class of "essential" subsemimodules (called here: semi- weakly-essential subsemimodules). In modules theory, it is well known that the congruence relations are defined by the sub- modules but not in theory of semimodules. So in the new class we consider the congruence relations defined only by the subsemimodules. We show that this new class contains the two known classes of "essential" subsemimodules. Futhermore, we study some interesting properties of this new class. As applications we introduce three notions of semi-weakly-co-hopfian semimodules. All semirings are associative with identity 1 (if R is a semiring, we assume that 1 6= 0), all semimodules are unital and all semiring extensions contain the common identity. Throughout this paper, for semimodule theoretic notions and notations we will follow [1] and [6]. In the following, we recall some definitions and notations that will be used in this paper. This work is organized as follows: • In section 1: Preliminaries: We give some results which we will use in the sequel. • In section 2: New notions of essential: Some properties of semi-weakly-essential sub- semimodules are investigated. • In section 3: Applications: Three types of semi-weakly-co-hopfian semimodules are in- troduced. 2. Preliminaries We recall briefly some basic notions about semimodules. E. Diop, D. Sow / Eur. J. Pure Appl. Math, 9 (2016), 250-265 252 We denote by N ≤ M , if N is a subsemimodule of a semimodule M and by homomorphism, we mean a homomorphism of left R-semimodules. Throughout this paper, we consider the left R-semimodules, but the results are also true for right R-semimodules and the proofs are similar. Definition 1. Let M be a left R-semimodule. • An R-congruence relation on a semimodule M is an equivalence relation ρ on M such that mρm′ and nρn′ =⇒ (m+ n)ρ(m′ + n′) and (rm)ρ(rm′), ∀m, m′, n, n′ ∈ M and r ∈ R. • The congruence relation ρ defined on M by mρm′⇐⇒ m= m′ is called a trivial congru- ence relation on M. • The congruence relation ρ defined on M by mρm′ ∀m, m′ ∈ M is called universal con- gruence relation on M. • M is a R-simple semimodule if any congruence relation defined over M is trivial or universal. Remark 1. The set of all R-congruence relation on M, R− cong(M), is partially-ordered by the relation ≤ defined by ρ ≤ ρ′ if and only if mρm′ =⇒ mρ′m′ ∀m, m′ ∈ M. For m, m′ ∈ M, ρ(m,m′) is the unique smallest element ρ of R− cong(M) satisfying mρm′. Definition 2. • A subsemimodule N of a semimodule M is called subtractive if for all m, m′ ∈ M, m+m′ ∈ N and m ∈ N implies m′ ∈ N. • The subtractive closure of a subsemimodule N of a semimodule M is the smallest subtractive subsemimodule of M containing N. • A semimodule M is said to be cancellative (additively cancellative) if for all m, m′, m′′ ∈ M, m+m′ = m+m′′ =⇒ m′ = m′′. Definition 3. Let N be a subsemimodule of a left R-semimodule M. N induces on M an R- congruence relation ≡N , know as the Bourne relation: ∀m, m′ ∈ M; m ≡N m′ ⇐⇒ ∃n, n′ ∈ N such that: m+ n= m′ + n′. • M/N denotes the factor R-semimodule M/ ≡N , and m/N denotes an element of M/N for some m ∈ M. • 0/N = N = {m ∈ M/∃n ∈ N/m+ n ∈ N} is the subtractive closure of N. Definition 4. Let M1 and M2 be subsemimodules of a left R-semimodule M. If M1 and M2 span M (i.e M = M1 + M2), and the restriction of ≡M2 to M1 and the restriction of ≡M1 to M2 are trivial, then M is the direct sum of its subsemimodules M1 and M2. And we write M = M1 ⊕M2. In this case for each m ∈ M, there exists unique pair (m1, m2) ∈ M1×M2 such that: m= m1+m2. In [6], we have the following characterization of essential subsemimodules. Notation: The class of essential subsemimodules of a left R-semimodule M is denoted by C RM . E. Diop, D. Sow / Eur. J. Pure Appl. Math, 9 (2016), 250-265 253 Lemma 1. If N is a subsemimodule of a left R-semimodule, M then the following conditions are equivalent: (i) N is essential (or large) in M; (ii) If ρ is a nontrivial R-congruence relation on M then the restriction of ρ to N is also non- trivial; (iii) If m and m′ are distinct elements of M then there exist distinct elements n and n′ of N satisfying nρ(m,m′)n ′. In [5], we have the following characterisation of semi-essential subsemimodules. Notation: The class of semi-essential subsemimodules of a left R-semimodule M is denoted by C RM . Lemma 2. A subsemimodule K of a left R-semimodule M is semi-essential if, and only if, for all x 6= 0, elements of M, there exists r ∈ R such that: 0 6= r x ∈ K. 3. New Notions of Essential: On Semi-Weakly-Essential Subsemimodules In [5] the class C RM of essential subsemimodules and the class C RM of semi-essential sub- semimodules were studied and neither one of these two classes is contained in the other. In this section these two classes are embedded in a new class namely the class wC RM of semi- weakly-essential subsemimodules. Definition 5. A subsemimodule N of a semimodule M is said to be semi-weakly-essential in M (denoted by N Ãswe M), if ∀h : M −→ M ′ R-homomorphism, the restriction of ≡Kerh to N is trivial =⇒ Kerh= 0. Proposition 1. If N is a subsemimodule of a left R-semimodule M then the following conditions are equivalent: (i) N is semi-weakly-essential in M; (ii) ∀K ≤ M, the restriction of ≡K to N is trivial⇒ K = 0. Proof. (i) =⇒ (ii). Let K ≤ M such that the restriction of ≡K to N is trivial. Let h : M −→ M/K be the surjection map. So Kerh= K . Since ≡K and ≡K are equivalent and the restriction of ≡K to N is trivial then restriction of ≡Kerh to N is trivial. N is semi-weakly-essential in M by assumption, so the restriction of ≡Kerh to N is trivial =⇒ Kerh= 0 i.e. K = 0 whence K = 0 is trivial. (ii) =⇒ (i). Indeed let h : M −→ M ′ an R-homomorphism such that the restriction of ≡Kerh to N is trivial. So by assumption we have the restriction of ≡Kerh to N is trivial =⇒ Kerh= 0. Proposition 2. If N is a subsemimodule of a left R-semimodule M, then we have: E. Diop, D. Sow / Eur. J. Pure Appl. Math, 9 (2016), 250-265 254 (i) N Ã M =⇒ N Ãswe M. (ii) N Ãs M =⇒ N Ãswe M. Proof. (i) N Ã M =⇒ N Ãswe M? Let h : M −→ M ′ an R-homomorphism such that the restriction of ≡Kerh to N is trivial. Suppose that ≡Kerh is nontrivial on M . So, we have an R-congruence relation on M namely ρ =≡Kerh which is nontrivial on M and whose restriction on N is trivial; con- tradiction because N Ã M . So ≡Kerh is trivial on M and we deduce that Kerh = 0, then N Ãswe M . (ii) N Ãs M =⇒ N Ãswe M? Let h : M −→ M ′ an R-homomorphism such that the restriction of ≡Kerh to N is trivial. So Kerh∩ N = 0 . Since N Ãs M , then Kerh∩ N = 0=⇒ Kerh= 0, then N Ãswe M . The previous proposition shows that the class of semi-weakly-essential subsemimodules of a semimodule M contains the two classes constituted by essential subsemimodules and semi-essential subsemimodules of M . Since R-congruence relations in modules theory are characterized by submodules, then the three notions of essential in semimodules theory, defined in this paper, are the same for modules theory. Recall that: Essential: ∀ρ ∈ R-Cong(M), ρ trivial on N =⇒ ρ trivial on M Semi-essential: ∀L ≤ M , L ∩ N = 0=⇒ L = 0 Semi-weakly essential: ∀L ≤ M , ≡L trivial on N =⇒ L = 0 Here we give some examples for the different notions of "essential" in semimodules theory. Example 1. In this example, we propose a finite semi-weakly-essential subsemimodule which is neither essential nor semi-essential. Set R= {0,1, . . . , n} (with n ∈ N and n≥ 2) and define on R the two commutative operations (⊕,⊗) as follows: (i) ∀x , y ∈ R : x ⊕ y =min(x , y) (ii) ∀x , y ∈ R : x ⊗ y =max(x , y). (R,⊕,⊗) is a semiring with 0R = n, 1R = 0. Let M ={0, 1 n , 1 n− 1 , 1 n− 2 , . . . 1 2 ,1,1+ 1 n , 1+ 1 n− 1 , E. Diop, D. Sow / Eur. J. Pure Appl. Math, 9 (2016), 250-265 255 . . . 1+ 1 2 ,2,2+ 1 n , . . . n− 1, (n− 1) + 1 n , . . . (n− 1) + 1 2 , n}.(M ,⊕) is a commutative monoid with 0M = n. Let "⋆" be the external operation defined by : ⋆ :R×M −→ M (r, m) 7−→ r ⋆m=max(r, m) It is clear that (M ,⊕,⋆) is an R-semimodule. Let a ∈ {0,1, . . . , n− 1}. Set Na = {a, . . . , n− 1 2 , n}; then Na is a subsemimodule of M. Now let us show that Na is semi-weakly-essential but is neither essential nor semi-essential. • Let us prove that Na is semi-weakly-essential. Let K ≤ M. Let us prove that the restriction of ≡K to Na is trivial if and only if K = {0M}. (=⇒) If K 6= {0M} then let k be an element of K such that k 6= 0M . Since k 6= n, we have [n− 1 2]⊕ k = n⊕ k, so n ≡K n− 1 2 with n 6= n− 1 2 , contradiction. Thus K = {0M}. (⇐=) Trivial. • Let us prove that Na is not semi-essential. Set L = {(n− 1) + 1 3 , n}. So L is a subsemimodule of M such that L 6= 0 and L ∩ Na = 0, then Na is not semi-essential. • Let us prove that Na is not semi-essential. Let ρ be the relation defined on M by: ∀(x , y) ∈ M×M, xρ y ⇐⇒max(x , 1) =max(y, 1). ρ is a congruence relation on M. Clearly ρ is not trivial on M and ρ is trivial on Na, hence Na is not essential. Example 2. In this example, we give an infinite semi-weakly-essential subsemimodule which is neither semi-essential nor essential. Recall the semiring R of the previous example. So M = [0, n] is a R-semimodule. Set Na = Q ∩ [a, n] where a ∈ Q∗+. By a same way as in previous example, we show that Na is semi-weakly-essential but is neither essential nor semi-essential. Example 3 ([5]). In this example, we propose a semi-essential subsemimodule which is a semi- weakly essential subsemimodule but not an essential subsemimodule. Let n ≥ 1 be an integer. Consider the set R = {r ∈ Q+/r ≤ n} ∪ {−∞} in which Q+ is the set of all nonnegative rational numbers, −∞ is assumed to satisfy the conditions that −∞ ≤ i and −∞ + i = −∞, ∀i ∈ R. Define on R the operations ⊕ and ⊗ as following: ∀i, j ∈ R; i ⊕ j =max(i; j) and i ⊗ j =min(i + j; n). We easily verify that (R;⊕;⊗) is a commutative semiring having −∞ as additive identity. R is also a left R-semimodule. Put R∗ = R\ {0}, then R∗ is an ideal of R. R∗ is a semi-essential subsemimodule of R, hence R∗ is a semi-weakly essential subsemimodule, but R∗ is not essential. E. Diop, D. Sow / Eur. J. Pure Appl. Math, 9 (2016), 250-265 256 Example 4 ([5]). In this example, we propose an essential subsemimodule which is a semi-weakly essential subsemimodule but not an semi-essential subsemimodule. Set R= {0,1, a} and define on R the two commutative operations (+,×) as following: (i) 0R = 0; 1R = 1 (ii) 1+ 1= 1+ a = 1; a+ 0= a+ a = a (iii) 0× 0= 0× 1= 0× a = 0; 1× 1= 1; 1× a = a× a = a. Then (R,+,×, 0, 1) is a commutative semiring. Let M = {0,1, a, b} with the same operations defined in R and 1M = 1R = 1, 0M = 0R = 0, b+ 0= b+ b = b, b+ 1= b+ a = a, 0× b = b× a = 0, b× 1= b× b = b. It’s easy to see that (M ,+,×, 0, 1) is a commutative R-semimodule. Now, put N = R= {0; 1; a}, then N is a semi-weakly essential subsemimodule of M, but N is not semi-essential. In the class of semi-weakly-essential subsemimodules, we have the following result: Proposition 3. Let M be a left R-semimodule, K and N be subsemimodules of M such that: K ≤ N ≤ M . Then we have: (i) K Ãs M ⇐⇒ (K Ãs N and N Ãs M). (ii) K Ã M ⇐⇒ (K Ã N and N Ã M). (iii) K Ãswe M =⇒ (K Ãswe N and N Ãswe M). Proof. (i) Similar methods to [5]. (ii) =⇒ Suppose that K Ã M . • Let n 6= n′ ∈ N . Since N ≤ M and K Ã M , there exist k 6= k′ ∈ K such that kρ(n,n′)k ′. Therefore K Ã N . • Let m 6= m′ ∈ M . Since K Ã M , there exist k 6= k′ ∈ K such that kρ(m,m′)k ′. But K ≤ N , so k, k′ ∈ N , therefore N Ã M . ⇐= Suppose that K Ã N and N Ã M . Let ρ be an R-congruence of M such that ρ is not trivial on M . Since N Ã M , we have the restriction of ρ to N is not trivial and since K Ã N , we have the restriction of ρ to K is not trivial, hence K Ã M . (iii) Trivial. E. Diop, D. Sow / Eur. J. Pure Appl. Math, 9 (2016), 250-265 257 Remark 2. We have the following remark. Let N ≤ K ≤ M and let ρ be and congruence relation which is trivial on K. So ρ is trivial on N but is not trivial in general on M. In the following lemma, we prove a particular case where we can enlarge a trivial congruence relation. Lemma 3. Let M be a left R-semimodule. Suppose that K1 ≤ M1 ≤ M ; K2 ≤ M2 ≤ M ; L1 ≤ M1; L2 ≤ M2 and M = M1 ⊕M2. Then: (i) ≡L1 trivial on K1 =⇒≡L1 trivial on K1 ⊕ K2. (ii) ≡L1⊕L2 trivial on K1 ⊕ K2 =⇒≡L1 trivial on K1 and ≡L2 trivial on K2. Proof. (i) Let k1 + k2, k′1 + k′2 be two elements of K1 ⊕ K2. We have k1 + k2 ≡L1 k′1 + k′2 =⇒ k1 + k2 + l1 = k′1 + k′2 + l ′1 =⇒ k1 + l1 ≡M2 k′1 + l ′1 and k2 ≡M1 k′2. Now by definition of M = M1 ⊕M2 and by hypothesis we have: k1 + l1 ≡M2 k′1 + l ′1 =⇒ k1 = k′1 and k2 ≡M1 k′2 =⇒ k2 = k′2. So k1 + k2 = k′1 + k′2 and therefore ≡L1 is trivial on K1 ⊕ K2. (ii) Suppose that ≡L1⊕L2 trivial on K1 ⊕ K2. Let k1, k′1 be two elements of K1 such that k1 ≡L1 k′1. So there exist l1, l ′1 ∈ L1 such that k1 + l1 = k′1 + l ′1. But k1 = k1 + 0 ∈ K1 ⊕ K2, l1 = l1 + 0 ∈ L1 ⊕ L2, idem we have k′1 ∈ K1 ⊕ K2, l1 ∈ L1 ⊕ L2. Since ≡L1⊕L2 trivial on K1 ⊕ K2, we deduce that k1 = k′1. Thus ≡L1 is trivial on K1 and a same way shows that ≡L2 is trivial on K2. Proposition 4. Let M be a left R-semimodule. Suppose that K1 ≤ M1 ≤ M; K2 ≤ M2 ≤ M and M = M1 ⊕M2. Then (K1 ⊕ K2)Ãswe (M1 ⊕M2) =⇒ (K1 Ãswe M1 and K2 Ãswe M2) Proof. Let us show that K1 Ãswe M1. Let L1 ≤ M1 such that ≡L1 is trivial on K1. By Lemma 3(i) we have ≡L1 trivial on K1 ⊕ K2. Now ≡L1 is trivial on K1 ⊕ K2 and K1 ⊕ K2 Ãswe M1 ⊕M2 so L1 = 0. Proposition 5. Let M be a left R-semimodule. Suppose that K1 ≤ M1 ≤ M; K2 ≤ M2 ≤ M and M = M1 ⊕M2. Then (K1 ⊕ K2)Ãs (M1 ⊕M2)⇐⇒ (K1 Ãs M1 and K2 Ãs M2). E. Diop, D. Sow / Eur. J. Pure Appl. Math, 9 (2016), 250-265 258 Proof. =⇒. Suppose for example K1 s M1. Then there exists a subsemimodule L1 6= 0 of M1 such that: L1 ∩ K1 = 0. So let us prove that L1 ∩ (K1 + K2) = 0. Let l1 ∈ L1 ∩ (K1 + K2). There exists (k1; k2) ∈ K1 × K2 such that: l1 = k1 + k2. We have: l1 ∈ L1 ≤ M1; k1 ∈ K1 ≤ M1 and k2 ∈ K2 ≤ M2, so by the direct sum M1 ⊕M2 we deduce that k2 = 0 and l1 = k1; hence l1 ∈ L1 ∩ K1 = 0, whence l1 = 0. Consequently L1 ∩ (K1 ⊕ K2) = 0 which contradicts the fact that (K1⊕K2)Ãs (M1⊕M2). So K1 Ãs M1. A same argument prove that K2 Ãs M2. ⇐=. Suppose that Ki Ãs Mi for all i ∈ {1,2}. Let 0 6= x ∈ M1 ⊕M2. Then, there exists (0,0) 6= (x1, x2) ∈ M1 ×M2 such that: 0 6= x = x1 + x2. Without lost of generality we can suppose that: 0 6= x1 ∈ M1. Since K1 Ãs M1 then from Lemma 1, there exists a r1 ∈ R such that: r1 x1 ∈ K1 and r1 x1 6= 0. • If r1 x2 ∈ K2 then r1 x1+r1 x2 ∈ K1+K2 therefore r1(x1+x2) ∈ K1⊕K2 with r1(x1+x2) 6= 0 because if r1 x1 + r1 x2 = 0, then by the sum direct we have r1 x1 = 0, which is absurd. Consequently (K1 ⊕ K2)Ãs (M1 ⊕M2). • If r1 x2 isn’t in K2 then there exists r2 ∈ R such that: 0 6= r2r1 x2 ∈ K2. We have r2r1 x1 ∈ K1 then r2r1(x1+ x2) ∈ K1⊕K2. If we put r = r2r1 then there exists r ∈ R such that: r(x1 + x2) ∈ K1 ⊕ K2 with r(x1 + x2) 6= 0 because if r x1 + r x2 = 0, then by the direct sum we have r x1 = 0, which is absurd. Therefore (K1 ⊕ K2) Ãs (M1 ⊕M2). Thus Ki Ãs Mi for all i ∈ {1; 2} =⇒ (K1 ⊕ K2)Ãs (M1 ⊕M2). Definition 6. Let N be a subsemimodule of a left R-semimodule M. A subsemimodule N ′ of M is called M-w-complement of N if the restriction of ≡N ′ to N is trivial and N ′ is maximal with this property. Proposition 6. (i) Every subsemimodule N of a left R-semimodule M has a M-w-complement. (ii) If N ′ is a M-w-complement of a subsemimodule N of M and if N ⊕ N ′ exists then: N ⊕ N ′ Ãswe M. Proof. (i) Let $ = {A≤ M/ the restriction of ≡A to N , is trivial}. 0 ∈ $, then $ 6= ∅. ($;⊆) is an ordered poset. It is easy to show that $ is a non-empty inductive poset. Therefore $ has at least one maximal element N ′. And N ′ is a M -ω-complement of N . (ii) • If N = 0 then N ′ = M , and so N ⊕ N ′ Ãswe M . • If N 6= 0, then let 0 6= L ≤ M such that the restriction of ≡L to N ⊕ N ′ is trivial. Let us show that ≡L is trivial on M . First of all we show that the restriction of ≡N ′+L to N is trivial. Let n1, n2 ∈ N such that n1 ≡N ′+L n2. E. Diop, D. Sow / Eur. J. Pure Appl. Math, 9 (2016), 250-265 259 n1 ≡N ′+L n2 =⇒ ∃n ′ 1, n′2 ∈ N , l1, l2 ∈ L such that n1 + n′1 + l1 = n2 + n′2 + l2. So by hypothesis and by the direct sum N ⊕ N ′ we deduce that n1 = n2. Therefore the restriction of ≡N ′+L to N is trivial. We deduce that N ′ + L ∈ $. Since N ′ is maximal, then N ′ + L = N ′ or N ′ + L = M . We have N ′ + L 6= M , otherwise, since the restriction of ≡N ′+L to N is trivial, we obtain (N ′ + L) ∩ N = M ∩ N = N = 0 which is a contradiction. So N ′ + L = N ′ and consequently L ⊆ N ′. Moreover we have (N⊕N ′)∩L = 0 because the restriction of≡L to N⊕N ′ is trivial; so N ′∩L = 0. Thus we have L ⊆ N ′ and N ′ ∩ L = 0, so L = 0 . We conclude that N ⊕ N ′ Ãswe M Definition 7. Let N be a subsemimodule of a left R-semimodule M. A subsemimodule N ′ of M is called M-s-complement of N if the restriction of ≡N to N ′ is trivial, the restriction of ≡N ′ to N is trivial and N ′ is maximal with this property. Proposition 7. (i) Every subsemimodule N of a left R-semimodule M has a M-s-complement. (ii) If M is a cancellative left R-semimodule and N a subsemimodule of M, then: (a) Every M-w-complement of N is a M-s-complement of N (b) If N ′ is a M-s-complement of N then: N ⊕ N ′ Ãswe M. Proof. (i). Similar to the above proof. For (2)(a) it suffices to see that, if M is cancellative and if the restriction of ≡N ′ to N is trivial, then the restriction of ≡N to N ′ is trivial. (2)(b). Similar to the above proof by using the fact that M is cancellative. 4. Applications: On Weakly Co-Hopfian Semimodules We have three differents notions of essential subsemimodules, so we can define three dif- ferents types of weakly co-hopfian semimodules 4.1. On Weakly Co-Hopfian Semimodules of Type 1 Definition 8. A nonzero left R-semimodule RM is said to be weakly co-hopfian-1 (denoted by wch-1) if every monomorphism f : M → M is semi-weakly-essential i.e f (M)Ãswe M. Proposition 8. The following are equivalent conditions on a left R-semimodule M. (i) M is weakly co-hopfian-1. (ii) ∀{0} 6= N ≤ M, if g ∈ End(M) is injective, then the restriction of≡N to g(M) is not trivial. E. Diop, D. Sow / Eur. J. Pure Appl. Math, 9 (2016), 250-265 260 Proof. (i)⇒(ii) Let {0} 6= N ≤ M and let g be an injective endomorphism of M . By assumption g(M) is semi-weakly-essential in M , so, since N ≤ M and N 6= {0}, we deduce that the restriction of ≡N to g(M) is not trivial. (ii)⇒(i) Let g : M → M be an injective endomorphism of M and let N be a subsemimodule of M such that the restriction of≡N to g(M) is trivial. Suppose that N 6= {0}. So by hypothesis the restriction of ≡N to g(M) is not trivial which contradicts the assumption. So N = {0} whence g(M) is semi-weakly-essential and so M is weakly co-hopfian-1. Proposition 9. For a left R-semimodule M, consider the following statements. (i) M is weakly co-hopfian-1. (ii) For any left R-semimodule N, if there is an R-monomorphism ˙M ⊕ N → M then N = {0}. Then (i) =⇒ (ii) and if M is cancellative we have (i)⇐⇒ (ii). NB: - A module M which verifies (ii) is said Dedekind finite. - A semimodule which is finite verifies (ii). - In the sequel, a semimodule M which verifies (ii) is said a F -semimodule. Proof. (i)⇒(ii) Suppose that f : ˙M ⊕ N → M is a monomorphism where N is a left R- semimodule. Let M i −→ ˙M ⊕ N f −→ M be an sequence where i is the canonical injection. Then f ◦ i is a monomorphism, hence ( f ◦ i)(M) is semi-weakly-essential in M by assumption. In an easy way one can show that f (N) = f ( ˙0⊕ N) = 0. Since f is monic, f (N) = 0=⇒ N = 0. (ii)⇒(i) By hypothesis we deduce this property (P): For any left R-semimodule N , if ˙M ⊕ N → M is an semi-weakly-essential monomorphism then N = {0}. Now let g : M → M be a monomorphism with non semi-weakly-essential image. Then, by Proposition 7 there exists a nonzero subsemimodule K with g(M)⊕ K Ãswe M . Define f : ˙M ⊕ K → M : (m, k) 7→ g(m) + k. By the direct sum g(M) ⊕ K and the fact that M is cancellative, f is monic. We have g(M) ⊕ K ⊆ f ( ˙M ⊕ K) ⊆ M , hence by Propo- sition 3 f ( ˙M ⊕ K) Ãswe M (because g(M) ⊕ K Ãswe M). So f is an semi-weakly-essential monomorphism contradicting the property (P) . Hence g(M)Ãswe M as desired. Proposition 10. The following are equivalent conditions on a left R-semimodule M. (i) M is weakly co-hopfian-1. (ii) M is a F-semimodule and the image of any injective endomorphism of M is either semi- weakly-essential or a proper direct summand. E. Diop, D. Sow / Eur. J. Pure Appl. Math, 9 (2016), 250-265 261 Proof. (i)⇒(ii) By hypothesis and by the previous proposition, we have: for any left R- semimodule N , if there is an R-monomorphism ˙M ⊕ N → M then N = {0}. Now suppose that ˙M ⊕ K ∼= M , then there is a monomorphism f : ˙M ⊕ K → M and we have K = {0}, hence M is a F -semimodule. By hypothesis the image of any injective endomorphism of M is in fact a semi-weakly-essential subsemimodule. (ii)⇒(i) Let g : M → M be an injective endomorphism and suppose that g(M) is not semi- weakly-essential in M . Then by hypothesis g(M) is a proper direct summand of M . So there exits a nonzero subsemimodule K of M such that g(M)⊕ K = M . Define f : ˙M ⊕ K → M : (m, k) 7→ g(m) + k. Then f is a monomorphism. We have M = g(M)⊕ K ⊆ f ( ˙M ⊕ K) ⊆ M , hence f ( ˙M ⊕ K) = M , whence f is surjective. Therefore there is an isomorphism ˙M ⊕ K ∼= M which contradicts the fact that M is a F - semimodule. Thus g(M)Ãswe M . Proposition 11. (i) The following are equivalent conditions on a left R-semimodule M. (a) M is weakly co-hopfian-1. (b) There exists a subsemimodule K of M such that g(K)Ãswe M for all injective g ∈ End(M). (ii) A direct summand of a weakly co-hopfian-1 semimodule is weakly co-hopfian-1. Proof. (i) (b)⇒(a) Trivial by Proposition 3. (a)⇒(b) Trivial. (ii) Suppose that M is a weakly co-hopfian-1 semimodule. Let M = N ⊕ K and let f : N −→ N be an injective endomorphism of N . Then the map f ⊕ IdK : M = N ⊕ K → M = N ⊕ K defined by n+ k 7→ ( f ⊕ IdK)(n+ k) = f (n) + k is an injective endomorphism of M and ( f ⊕ IdK)(M) Ãswe M . Then ( f ⊕ IdK)(M) Ãswe M ⇒ f (N) ⊕ K Ãswe N ⊕ K and by Proposition 4 f (N)Ãswe N , therefore N is weakly co-hopfian-1. 4.2. On Weakly Co-Hopfian Semimodules of Type 2 Definition 9. A nonzero left R-semimodule RM is said to be weakly co-hopfian-2 (denoted by wch-2) if every monomorphism f : M → M is semi-essential i.e f (M)Ãs M (see definition in the introduction). E. Diop, D. Sow / Eur. J. Pure Appl. Math, 9 (2016), 250-265 262 Proposition 12. If M is wch-2, then M is wch-1. Proof. By the Proposition 2. Proposition 13. The following are equivalent conditions on a left R-semimodule M. (i) M is weakly co-hopfian-2. (ii) For every monomorphism f : M → M, if x is an nonzero element of M, then there exists r ∈ R such that 0 6= r x ∈ f (M). (iii) Injective endomorphisms of M map semi-essential subsemimodules to semi-essential sub- semimodules. Proof. (i)⇐⇒(ii) By Definition 9 and Lemma 2. (iii)⇐⇒(i) is trivial. (i)⇐⇒(iii) Let g : M → M be an injective endomorphism of M, and let K be a semi-essential subsemimodule of M. Let us prove that g(K)Ãs M. We have g(K)≤ g(M)≤ M and by hypothesis g(M)Ãs M, so, according to the Proposition 3, to obtain g(K)Ãs M, it suffices to prove that g(K)Ãs g(M). Let x ∈ g(M). So there exists m ∈ M such that x = g(m). Since K Ãs M, we deduce that there exists an r ∈ R such that rm ∈ K. So g(rm) = r g(m) = r x ∈ g(K); thus g(K)Ãs g(M). Proposition 14. (i) A direct summand of a weakly co-hopfian-2 semimodule is weakly co-hopfian-2. (ii) Let M = M1 ⊕ M2 such that each Mi is fully invariant. Then M is weakly co-hopfian-2 if and only if so is each Mi . Proof. (i) Suppose that M is a weakly co-hopfian-2 semimodule. Let M = N ⊕ K and let f : N −→ N be an injective endomorphism of N . Then the map f ⊕ IdK : M = N ⊕ K → M = N ⊕ K defined by n + k 7→ ( f ⊕ IdK)(n + k) = f (n) + k is an injective endomorphism of M and so ( f ⊕ IdK)(M) Ãs M . Since ( f ⊕ IdK)(M) ≤ f (N)⊕ K , we deduce by Proposition 3 that f (N) ⊕ K Ãs N ⊕ K and by Proposition 5 that f (N) Ãs N , therefore N is weakly co-hopfian-2. (2)=⇒) By (i). ⇐=) Let f : M = M1⊕M2 −→ M = M1⊕M2 be an injective endomorphism of M = M1⊕M2. We have f (M1)⊕ f (M2) ⊆ f (M1 ⊕M2) ⊆ M1 ⊕M2. By assumption we have f (M1) Ãs M1 and f (M2) Ãs M2, and by the Proposition 5, we obtain f (M1) ⊕ f (M2) Ãs M1 ⊕ M2 and consequently f (M1 ⊕M2)Ãs M1 ⊕M2. E. Diop, D. Sow / Eur. J. Pure Appl. Math, 9 (2016), 250-265 263 4.3. On Weakly Co-Hopfian Semimodules of Type 3 Definition 10. A nonzero left R-semimodule RM is said to be weakly co-hopfian-3 (denoted by wch-3) if every monomorphism f : M → M is essential i.e f (M)Ã M. Proposition 15. The following are equivalent conditions on a left R-semimodule M. (i) M is weakly co-hopfian-3. (ii) If m and m′ are distinct elements of M then for every monomorphism f : M → M, there exist distinct elements f (m1) and f (m2) of f (M) satisfying f (m1)ρ(m,m′) f (m2). Proof. By Definition 10 and Lemma 1. Proposition 16. If M is wch-3, then M is wch-1. Proof. By Proposition 2. By a same way as above, we show the following results. Proposition 17. (i) The following are equivalent conditions on a left R-semimodule M. (a) M is weakly co-hopfian-3. (b) There exists a subsemimodule K of M such that g(K)Ã M for all injective g ∈ End(M). (ii) A direct summand of a weakly co-hopfian-3 semimodule is weakly co-hopfian-3. (iii) Let M = M1 ⊕ M2 such that each Mi is fully invariant.Then M is weakly co-hopfian-3 if and only if so is each Mi . Proposition 18. Any R−simple semimodule is wch-3 Proof. Obvious. 4.4. Examples of Co-Hopfian Semimodules Here we give examples of co-hopfian semimodules. Example 5. Recall the semimodule (M ,⊕,⋆) of Example 1. Let us show that M is wch-1, wch-2 and wch-3. Let g : M −→ M be an injective endomorphism of M. So g verifies the following property: ∀m, m′ ∈ M: m≤ m′ =⇒ g(m)≤ g(m′). This property with the injectivity of g make that there exists an unique injective endomorphism of M, namely the identity map on M. 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