E extracta mathematicae Vol. 32, N´um.2, 239 { 254 (2017) Content Semimodules Rafieh Razavi Nazari, Shaban Ghalandarzadeh Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran [email protected] [email protected] Presented by Juan Antonio Navarro Received November 27, 2016 Abstract: The purpose of this paper is to study content semimodules. We obtain some results on content semimodules similar to the corresponding ones on content modules. We study normally flat content semimodules and multiplication content semimodules. More- over, we characterize content semimodules over discrete valuation semirings and Boolean algebras. Key words: Semiring, Content semimodule, Multiplication semimodule, Normally flat semi- module. AMS Subject Class. (2010): 16Y60. Introduction Semirings and semimodules have many applications in different branches of mathematics (see [7], [8] and [9]). Semiring is a generalization of ring and bounded distributive lattice. We recall here some definitions: A semiring is a nonempty set S with two binary operations addition (+) and multiplication (·) such that the following conditions hold: 1) (S; +) is a commutative monoid with identity element 0; 2) (S; :) is a monoid with identity element 1 =6 0; 3) 0a = 0 = a0 for all a 2 S; 4) a(b + c) = ab + ac and (b + c)a = ba + ca for every a; b; c 2 S. The semiring S is commutative if the monoid (S; :) is commutative. All semirings in this paper are commutative. An ideal I of a semiring S is a nonempty subset of S such that a + b 2 I and sa 2 I for all a; b 2 I and s 2 S. An ideal I is subtractive if a + b 2 I and b 2 I imply that a 2 I for all a; b 2 S. A semiring is entire if ab = 0 implies that a = 0 or b = 0. Further, an element a of a semiring S is multiplicatively cancellable (abbreviated as MC) if ab = ac implies that b = c. If every nonzero element of S is multiplicatively 239 240 r. razavi nazari, sh. ghalandarzadeh cancellable we say that the semiring S is a semidomain. An element a of a semiring S is multiplicatively idempotent if a2 = a. Let I×(S) denote the set of all multiplicatively idempotent elements of S. We say that S is multiplicatively idempotent, if I×(S) = S. Let S be a semiring. An S-semimodule is an additive abelian monoid (M; +) with additive identity 0M and a function S × M ! M ((s; m) 7! sm), called scalar multiplication, such that the following conditions hold for all s; s0 2 S and all m; m0 2 M: 1) (ss0)m = s(s0m); 2) s(m + m0) = sm + sm0; 3) (s + s0)m = sm + s0m; 4) 1m = m; 5) s0M = 0M = 0m; A subset N of an S-semimodule M is a subsemimodule of M if N is closed under addition and scalar multiplication. We say that a subsemimodule N of an S-semimodule M is subtractive if m+m0 2 N and m 2 N imply that m0 2 N for all m; m0 2 M. Let M and M 0 be S-semimodules. Then a function α from M to M 0 is an S-homomorphism if α(m + m0) = α(m) + α(m0) for all m; m0 2 M and α(sm) = s(α(m)) for all m 2 M and s 2 S. The kernel of α is ker(α) = α−1f0g. Then ker(α) is a subtractive S-semimodule of M. The set α(M) = fα(m) j m 2 Mg is a subsemimodule of M 0. An S-homomorphism α : M ! M 0 is an S- monomorphism if αβ = αβ0 implies β = β0 for all S-semimodule K and all S-homomorphisms β; β0 : K ! M. If α is an S-monomorphism, then ker(α) = 0. But the converse need not be true. For example, let S be an entire semiring and b 2 S such that it is not multiplicatively cancellative. Thus there exists a =6 a0 2 S such that ab = a0b. Define a map ϕ : S ! Sb by s 7! sb. Then ϕ is an S-homomorphism with ker(ϕ) = 0. But ϕ is not injective, since ϕ(a) = ϕ(a0). An S-homomorphism α : M ! M 0 is surjective if α(M) = M 0. Let S be a semiring and M an S-semimodule. For any x 2 M, we de- fine c(x) = \fI j I is an ideal of S and x 2 IMg. Then c is a function from M to the set of ideals of S and it is called the content function. An S- semimodule M is called a content semimodule if for every x 2 M, x 2 c(x)M. In this paper, we study content semimodules and extend some results of [14] to semimodules over semirings. In Section 1, we recall some properties of content semimodules 241 content semimodules from [12] and we show that projective semimodules are content semimodules. We study normally flat content semimodules in Sec- tion 2. In Section 3, we characterize content S-semimodules over discrete valuation semirings. In Section 4, we investigate some properties of faithful multiplication content semimodules, as a generalization of faithful multipli- cation modules. In the last section, we prove that if every subsemimodule of a content S-semimodule is a content S-semimodule with restricted content function, then S is a multiplicatively regular semiring. We also characterize content semimodules over Boolean algebras. 1. Content semimodules The concepts of content modules and content algebras were introduced in [14]. The concept of content semimodules is studied in [12]. Let S be a semiring and M an S-semimodule. For any x 2 M, we define the content of x by, { } cS;M (x) = \ I j I is an ideal of S and x 2 IM : Therefore cS;M is a function from M to the set of ideals of S which is called the content function.P If N is any non-empty subset of M, we define cS;M (N) to be the ideal x2N cS;M (x). Whenever there is no fear of ambiguity, either or both of the subscripts S and M will be omitted. Definition 1. Let S be a semiring. An S-semimodule M will be called a content S-semimodule if for every x 2 M, x 2 c(x)M. Example 2. Let S be a multiplicatively idempotent semiring, J an ideal of S and x 2 J. It is clear that (x) ⊆ c(x) = \fI j I ⊆ S; x 2 IJg. Thus (x)J ⊆ c(x)J. But x = x2 2 (x)J and hence x 2 c(x)J. Therefore J is a content S-semimodule. Now, we recall next results from [12], which will be used repeatedly. Theorem 3. Let M be an S-semimodule. Then the following statements are equivalent: 1) M is a content S-semimodule. 2) For every set of ideals fIig of S, (\Ii)M = \(IiM). 3) For every set of finitely generated ideals fIig of S, (\Ii)M = \(IiM). 242 r. razavi nazari, sh. ghalandarzadeh 4) There exists a function f from M to the set of ideals of S such that for every x 2 M and every ideal I of S, x 2 IM if and only if f(x) ⊆ I. Moreover, if M is a content S-semimodule and x 2 M, then c(x) is a finitely generated ideal. Theorem 4. Let M be a content S-semimodule, and N a subsemimodule of M. Then the following statements are equivalent: 1) IM \ N = IN for every ideal I of S. 2) For every x 2 N, x 2 cM (x)N. 3) N is a content S-semimodule and cM restricted to N is cN . We know that every free module and every projective module as a direct summand of a free module, are content modules by [14, Corollary 1.4]. More- over every free semimodule is a content semimodule by [12, Corollary 26]. But not all projective semimodules are direct summands of free semimodules (cf. [3, Example 2.3]). We can prove that every projective semimodule is a content semimodule as follows: Theorem 5. Any projective semimodule is a content semimodule. Proof. Let S be a semiring and M a projective S-semimodule. Then by [18, f g ⊆ f g ⊆ Theorem 3.4.12], there exist mi i2I M and fi i2I HomP S(M; S) such 2 2 that for any x M, fi(x) =P 0 for almost all i I, and x = i fi(x)mi. Sup- 2 n 2 pose that x M. Then x = i=1 fi(x)mi and hence x (f1(x); : : : ; fn(x))M. Thus c(x) ⊆ (f1(x); : : : ; fn(x)). Now assume that x 2 IM for some ideal I 2 N 2 2 of S. ThenP there exist m , r1; : : : ; rm I andP x1; : : : ; xm M such m ≤ ≤ m that x = i=1 rixi. For each 1 j n, fj(x) = i=1 rifj(xi), and hence fj(x) 2 (r1; : : : ; rm). Therefore (f1(x); : : : ; fn(x)) ⊆ (r1; : : : ; rm) ⊆ I. This implies (f1(x); : : : ; fn(x)) ⊆ c(x), by definition of content function. Therefore (f1(x); : : : ; fn(x)) = c(x), and x 2 (f1(x); : : : ; fn(x))M = c(x)M. Let M be an S-semimodule, N a subsemimodule of M and I an ideal of S. Put (N :M I) = fx j x 2 M and Ix ⊆ Ng: Then (N :M I) is a subsemimodule of M.