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International Journal of Algebra, Vol. 4, 2010, no. 26, 1299 - 1306

Prime Subsemimodules of Semimodules

Reza Ebrahimi Atani Department of Computer Engineering University of Guilan P.O. Box 3756, Rasht, Iran [email protected]

Abstract In this paper we characterize prime subsemimodules of a semimodule over a commutative semiring.

Mathematics Subject classification: 16Y60 Keywords: Quotient semimodules, Strong ideals, Prime subsemimodules

1

Introduction

Semimodules over semirings also appear naturally in many areas of mathematics. For example, semimodules are useful in the area of theoretical computer science as well as in the cryptography [10]. In the present paper, we introduce and investigate the quotient semimodules of a module over a semiring. Moreover, we extend some basic results of Lu [9] to semimodules over semirings. For the definitions of monoid, semirings, semimodules and subsemimodules of a semimodule we refer [8, 6, 1, 2]. All semiring in this paper are commutative with non-zero identity. A semiring R is said to be semidomain whenever a, b R with ab = 0 implies that either a = 0 or b = 0. A semifield is a semiring in which non-zero elements form a group under multiplication. The semiring R is considered to be also a semimodule over itself. In this case, the subsemimodules of R are called ideals of R. An R-semimodule M is said to be semivector space if R is a semifield. Let M be a semimodule over a semiring R. A subtractive subsemimodule (= k-subsemimodule) N is a subsemimodule of M such that if x, x + y N, then y N (so {0M } is a k-subsemimodule of M). A prime subsemimodule of M is a proper subsemimodule N of M in which x N or rM N whenever rx N. We define k-ideals and prime ideals of a semiring R in a similar fashion. We say that r R is a zero-divosor for a semimodule M if rm = 0 for some non-zero element m of M. The set of zero-divisors of M is written ZR (M).

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Quotient semimodules

Quotient semimodules are determined by equivalence relations rather than by subsemimodules as in the module case. If N is a sunsemimodule of a semimodule M over a semiring R, we define a relation on M, given by m1 m2 if and only if there exist n1 , n2 N satisfying m1 + n1 = m2 + n2 . Then is an equivalence relation on M, and we denote the equivalence class of m by m + N and these collection of all equivalence classes by M/N. Then M/N forms a commutative additive semigroup which has zero element under the binary operation defined as follows: (m + N) (m + N) = m + m + N. Then 0M + N is the zero element of M/N. Now let r R and suppose that m1 + N, m2 + N M/N are such that m1 + N = m2 + N in M/N. Then there are elements a, b N such that rm1 + ra = rm2 + rb, so rm1 rm2 ; hence rm1 + N = rm2 + N. Hence we can unambiguously define a mapping from R × M/N into M/N (sending (r, m + N) to rm + N) and it is routine to check that this turns the commutative semigroup M/N into an R-semimodule. We call this R-semimodule the quotient semimodule or factor semimodule of M modulo N. Lemma 2.1 Let N be a subsemimodule over a semimodule M over a semiring R. Then the following hold: (i) If a N, then a + N = N. (ii) If N is a k-subsemimodule of M and a N, then a + N = b + N for every b M if and only if b N. In particular, c + N = N if and only if c N. Proof. (i) Since a + 0M = 0M + a, we conclude that a 0; hence a + N = 0 + N = N. (ii) Let a + N = b + N for every b M. Then a + u = b + v for some u, v N; so b N since N is a k-subsemimodule. The other implication follows from (i) and the fact that N is a k-subsemimodule of M. 2 Proposition 2.2 Let N and K be subsemimodules of a semimodule M over a semiring R with N K. Then K/N = {m + N : m K} is a subsemimodule of M/N. In particular, if K is a k-subsemimodule of M, then K/N is a k-subsemimodule of M/N. Proof. Clearly, 0 + N K/N. Let m + N, m + N K/N and r R. It is easy to see that (m + N) (m + N) = m + m + N K/N and r(m + N) = rm + N K/N. Thus K/N is a subsemimodule of M/N. Finally, assume that u + N K/N and (u + N) (v + N) = u + v + N K/N, where u K and v M. It then follows that u + v + t1 = c + t2 for some t1 , t2 N and c K; hence v K since K is a k-subsemimodule. Thus v + N K/N, and the proof is complete. 2

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Theorem 2.3 Let M be a semimodule over a semiring R, N a subsemimodule of M and L a k-subsemimodule of M/N. Then L = T /N for some ksubsemimodule of T of M. Proof. Assume that T = {m M : m + N L} and let n N. Then by Lemma 2.1, n + N = 0M + N L; hence N T . Let a, b T and r R. Then (a + N) (b + N) = a + b + N L; so a + b T . Similarly, ra T . Thus T is a subsemimodule of M. Let a, a + b T . Then a + N, a + b + N = (a + N) (b + N) L; hence b + N L since L is a k-subsemimodule of M/N. It follows that b T , whence T is a k-subsemimodule of M. Finally, an inspection will show that L = T /N. 2 Let M and N be semimodules over the semiring R, and let f : M N be a homomorphism of R-semimodules. It is easy to see that Kerf = {m M : f (m) = 0N } is a subsemimodule of M and Im(f ) is a subsemimodule of N. Theorem 2.4 Let M and N be semimodules over the semiring R, and let f : M N be a homomorphism of R-semimodules. Then f induces an ¯ ¯ isomorphism f : M/Kerf Imf for which f (m + Kerf ) = f (m) for all m M. In particular, if f is surjective, then M/Kerf N. = Proof. The proof is straightforward. 2

Let R be a semiring. We define the Jacobson radical of R, denoted by Jac(R), to be the intersection of all the maximal k-ideals of R. Then by [3, Lemma 2], the Jacobson radical of R always exists and by [2, Lemma 2.12], it is a k-ideal of R. A non-zero element a of R is said to be semi-unit in R if there exist r, s R such that 1 + ra = sa. An proper ideal I of a semiring R is said to be a strong ideal, if for each a I there exists b I such that a + b = 0 (see [5]) . Now we state and prove a version of Nakayama's lemma. Theorem 2.5 Let M be a finitely generated semimodule over a semiring R and let J be a strong ideal of R contained in the Jacobson radical of R. Then the following hold: (i) If M = JM, then M = 0. (ii) If N is a subsemimodule of M such that JM + N = M, then M = N. Proof. By [6, Lemma 2.2], (1 + t)M = 0 for some t J. Let x M. Then (1 + t)x = 0. Since 1 + t is a semi-unit by [7, Lemma 3.4], we must have 1 + (1 + t)t = (1 + t)s for some t, s R; hence x = 0. (ii) Let m1 , m2 , ..., mk be elements which generate M. Then the R-semimodule generated by m1 + N, m2 + N, ..., mk + N is just M/N. Since J(M/N) = (IM + N)/N = M/N, we must have M/N = {0 + N} by (i). Now the assertion follows from Lemma 2.1. 2

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Basic properties of prime subsemimodules

Let R be a semiring, M an R-semimodule and N a prime subsemimodules of M. Then (N :R M) is a prime ideal of R [4, Lemma 9]. Let M be a semimodule over a semiring R. We say that M is a torsion-free R-semimodule whenever r R and m M with rm = 0 implies that either m = 0 or r = 0 (so every semivector space over a semifield R is a torsion-free R-semimodule). Allen [1] has presented the notion of a Q-ideal I in the semiring R and constructed the quotient semiring R/I. The results proven in [1] and [2] will be used in the next results. Remark 3.1 (Change of semirings) Assume that I is a Q-ideal of a semiring R and let N be a subsemimodule of an R-semimodule M. We show now how M/N can be given a natural structure as a semimodule over R/I. Let q1 , q2 Q such that q1 + I = q2 + I, and let m, m M such that m + N = m + N. Then q1 = q2 , and q1 m + q1 n = q2 m + q1 n for some n, n N; thus q1 m + N = q2 m + N. Hence we can unambiguously define a mapping R/I × M/N into M/N (sending (q1 + I, m + N) to q1 m + N) and it is routine to check that this turns the commutative additive semigroup with a zero element M/N into an R/I-semimodule. It should be noted that a subset of M/N is an Rsubsemimodule if and only if it is an R/I-subsemimodule. We next give six other characterizations of prime subsemimodules. Theorem 3.2 Let N be a proper k-subsemimodule of a semimodule over a semiring R with (N : M) = P a Q-ideal of R. Then the following statements are equivalent: (1) N is a prime subsemimodule of M; (2) M/N is a torsion-free R/P -semimodule; (3) (N :M < r >) = N for every r R - P ; (4) (N :M J) = N for every ideal J P ; (5) (N :R < m >) = P for every m M - N; (6) (N :R L) = P for every subsemimodule L of M properly containing N; (7) ZR (M/N) = P . Proof. (1) (2) Note that M/N is an R/P -semimodule by Remark 3.1. Let (q + P )(m + N) = qm + N = 0M + N where q Q and m M, so qm N by Lemma 2.1. Therefore, N prime gives either q P or m N. If q P , then q + P is the zero in R/P (otherwise, m + N is the zero in M/N by Lemma 2.1 again). Thus M/N is torsion-free semimodule as an R/P -semimodule. (2) (3) Assume that q0 + P is the zero element in R/P . It suffices to show that (N :M < r >) N. Let m (N :M < r >). Then rm N and

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r = q + a for some q Q and a P (so q P ); hence qm N since N / is a k-subsemimodule. Since (q + P )(m + N) = 0M + N by Lemma 2.1 and q + P = q0 + P , we must have m + N = 0M + N; hence m N by Lemma 2.1, and so we have equality. (3) (4) Clearly, N (N :M J). For the reverse inclusion, assume that m (N :M J). By assumption, there exist r J such that r R - P and rm N; so (N :M < r >) = N by (3). This completes the proof. (4) (5) Since P M N, we conclude that P (N :R < m >) for every m M - N. for the other containment, assume that m M - N and r (N :R < m >); we show that r P . Suppose not. Then J =< r > P , and so m (N :M J) = N by (4), which is a contradiction, as required. (5) (6) If a P , then aL aM N; so P (N :R L). Now suppose that b (N :R L). By assumption, there exists m L such that m M - N. Then b (N :R < m >) = P by (5), as needed. (6) (7) Let r ZR (M/N). Then there exists m M - N such that r(m + N) = rm + N = 0M + N, so rm N by Lemma 2.1; hence r (N :R Rm + N) = P by (6). Thus ZR (M/N) P . For the reverse conclusion, assume that a P . By assumption, there is an element m M - N such that am N, so a(m + N) = am + N = 0M + N; thus a ZR (M/N). This completes the proof. (7) (1) Let rm N for some r R and m M - N; we show that r P . Then r(m + N) = 0M + N by Lemma 2.1; hence r ZR (M/N) = P by (7), as required. 2 Proposition 3.3 Let N be a proper k-subsemimodule of a semimodule M over a semiring R with (N : M) = P a maximal Q-ideal of R. Then N is a prime subsemimodule. In particular, P M is a prime subsemimodule of an R-semimodule M for every maximal Q-ideal P of R such that P M = M. Proof. By [2, Theorem 2.10], R/P is a semifield, so M/N is a semivector space over the semifield R/P by Remark 3.1; hence it is a torsion-free R/P semimodule. Thus N is prime by Theorem 3.2. Finally, suppose that (P M : M) = J = R. Then P J, so J = P since P is maximal, as required. 2 Proposition 3.4 Let N be a proper k-subsemimodule of a semimodule M over a semiring R with (N : M) = P a Q-ideal of R and let P be a maximal ideal of R. Then N is a P -prime if and only if P M N. In particular, if N is an P -prime subsemimodule of M, then so is every proper subsemimodule of M containing N. Proof. It suffices to show that if P M M, then N is P -prime. Let p P . Then p (N : M), so P = (N : M) by maximality of P . Now apply Proposition 3.3. 2

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Proposition 3.5 Let N be a proper maximal k-subsemimodule of a semimodule over a semiring R. Then N is a prime subsemimodule and (N : M) is a maximal ideal of R. Proof. Let rm N such that r R and m M - N. Note that N is a maximal subsemimodule if and only if M/N is a simple R-semimodule. Hence M/N is the cyclic R-semimodule Rm, where m = m+N. By assumption and Lemma ¯ ¯ 2.1, we must have r m = 0M + N, so r(M/N) = r(Rm) = R(r m) = 0M + N; ¯ ¯ ¯ hence rM N. Thus N is prime. Finally, The mapping f : R Rm defined ¯ by f (r) = r m for all r R is a surjective R-semimodule homomorphism, so ¯ R/Kerf Rm by Theorem 2.4; hence Kerf = ann(m) = ann(M/N) = (N : ¯ = ¯ M) is a maximal ideal of R. 2 Proposition 3.6 Let N1 , N2 , ..., Nn be subsemimodules of an R-semimodule M and let N be a prime subsemimodule of M. If N1 N2 ... Nn N, then there exists an i such that either Ni N or (Ni : M) (N : M). / Proof. Suppose not. Then there exists m N1 such that m N and a / ai (Ni : M) such that ai (N : M) for every i = 1. Therefore, ai m N1 Ni for every i = 1 so that a2 a3 ...an m N1 N2 ... Nn N. However, m N / / and a2 a3 ...an (N : M), which is a contradiction. 2 Lemma 3.7 Let R be a semiring, I a strong ideal in R, M an R-semimodule generated by n elements, and x an element of R satisfying xM IM. Then (xn + y)M = 0 for some y I. Proof. We use induction on n. Consider first the case in which n = 1. Here we have x < m > I < m >. So xm = sm for some s I; hence there is an element s I such that (x + s )m = sm + s m = 0. It follows that (x + s )M = 0. We now turn to the inductive step. Assume, inductively, that n = k + 1, where k 1, and that the result has been proved in the case where n = k. Then we must have (x + a)(xk + b)M = (xk+1 + axk + bx + ab)(< m1 , ..., mk > + < mk+1 >) = 0 for some a, b I, so (xk+1 + c)M = 0, where axk + bx + ab = c J. This completes the proof. 2 Proposition 3.8 Let M be a finitely generated semimodule over a semiring R and let I be a strong k-ideal of R such that I = rad(I). Then (IM : M) = I if and only if ann(M) I. Proof. The necessity is clear. Assume that ann(M) I and let x (IM : M). If M generated by n elements, then there exists a y I such that xn + y ann(M) I by Lemma 3.7. Since I is a k-ideal, we must have xn I and, therefore, (IM : M) rad(I) = I. Now we can see easily that (IM : M) = I. 2

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Theorem 3.9 If M is a finitely generated semimodule over a semiring R and P is a maximal Q-ideal of R containing ann(M), then P M = M so that P M is a prime subsemimodule of M. In particular, If M is a finitely generated faithful R-semimodule, then P M is a prime subsemimodule of M for every maximal Q-ideal P of R. Proof. Apply Proposition 3.3 and Proposition 3.8 (not that every Q-ideal is a k-ideal). 2 Acknowledgements I would like to thank Prof. S. Ebrahimi Atani for several useful suggestions on the first draft of the manuscript.

References

[1] P. J. Allen, A fundamental theorem of homomorphisms for simirings, Proc. Amer. Math. Soc. 21 (1969), 412-416. [2] S. Ebrahimi Atani, The ideal theory in quotients of commutative semirings, Glasnik Matematicki 42 (2007), 301-308. [3] R. Ebrahimi Atani and S. Ebrahimi Atani, Ideal theory in commutative semirings, Bul. Acad. Stiinte Repub. Mold. Mat 2 (2008), 14-23. [4] R. Ebrahimi Atani and S. Ebrahimi Atani, On subsemimodules of semimodules, Bul. Acad. Stiinte Repub. Mold. Mat. to appear (2010). [5] S. Ebrahimi Atani and R. Ebrahimi Atani, Some remarks on partitioning semirings, An. St. Univ. Ovidius Constanta, 18 (1) (2010), 49-62. [6] S. Ebrahimi Atani and M. Shajari Kohan, A note on finitely generated multiplication semimodules over comutative semirings, International Journal of Algebra 4(8) (2010), 389-396. [7] S. Ebrahimi Atani, The zero-divisor graph with respect to ideals of a commutative semiring, Glas. Math. 43 (2008), 309-320. [8] J. S. Golan, The theory of semirings with applications in mathematics and theoretical computer Science, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific and Technical, Harlow UK, 1992. [9] C. P. Lu, Prime submodules of modules. Comment. Univ. St. Pauli 33 (1984), 61-69.

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[10] G. Maze, C. Monico and J. Rosenthal, Public key cryptography based on semigroup actions, Adv. Mathematics of communication, 1(4) (2007), 489-281. Received: July, 2010

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