Microsoft Word - 102-108 Mathematics | 102 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.1999 Vol. 31 (3) 2018 For Some Results of Semisecond Submodules Rasha I. Khalaf Department, of Mathematics, College of Education for Pure Science Ibn Al-Haitham, University of Baghdad, Baghdad, Iraq. rasha_sin79@yahoo.com Article history: Received 15 May 2018, Accepted 19 September 2018, Published December 2018 Abstract Let ℛ be a commutative ring with unity and let ℬ be a unitary R-module. Let ℵ be a proper submodule of ℬ, ℵ is called semisecond submodule if for any r∈ℛ, r≠0, n∈Z+, either rnℵ=0 or rnℵ=rℵ. In this work, we introduce the concept of semisecond submodule and confer numerous properties concerning with this notion. Also we study semisecond modules as a popularization of second modules, where an ℛ-module ℬ is called semisecond, if ℬ is semisecond submodul of ℬ. Keywords: Semisecond submodules, second submodules, secondary submodules. 1. Introduction Let ℛ be a commutative ring with unity and let ℬ be a unitary ℛ -module. S.Yass in [1] introduced the notation of second submodule and second module where a submodule ℵ of an ℛ -module ℬ is called second submodule if for every r∈ℛ, r≠0, either rℵ =ℵ or rℵ =0 and a module ℬ is called semisecond if ℬ is semisecond submodule of ℬ. This definition leads us to introduce the notion of semisecond submodule and semisecond module as a generalization of second submodule and second module, where a submodule ℵ of an ℛ-module ℬ is called Semisecond if for every r∈ ℛ, r≠0, n∈Z+, either rnℵ =0 or rnℵ =rℵ and a module ℬ is Semisecond if ℬ is semisecond submodule of ℬ. The main aim of this work is to give basic properties of Semisecond submodules. Moreover, we survey the relationships between semisecond submodules and other submodules. Over this work we designate S.R.M. for submodule of an ℛ-module, for integral domain, for finitely generated, s.t. for such that and N.Z. for non-zero. 2. Semisecond Submodules Definition (1):-let ℵ be a S.R.M. ℬ, ℵ is semisecond submodule if for every r∈ℛ, n∈Z+, either rnℵ=0 or rnℵ=rℵ. An ideal I of a ring ℛ is semisecond ideal if it is semisecond submodule of the ℛ -module ℛ. The later result is a description of semisecond submodule. Proposition (2):- ℵ is S.R.M. ℬ is semisecond iff r2 ℵ =0 or r2 ℵ =r ℵ for any r∈ ℛ, r≠0. Proof:-(⟹) Is obvious. (⟸) if r=3, then r3ℵ =r(r2ℵ). Since either r2 ℵ =0 or r2 ℵ =r ℵ, that is either r3ℵ =r(0)=0 or r3ℵ =r(rℵ) =r2ℵ =rℵ. Suppose that rn ℵ =0 or rn ℵ = r ℵ is whole for n=k. To evidence that the permit is whole if n=k+1. (r)k+1 ℵ =r(rk ℵ). But rk ℵ =0 or rkℵ =rℵ, that is rk+1ℵ =r(0) =0 or Mathematics | 103 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.1999 Vol. 31 (3) 2018 (r)k+1 ℵ =r(r ℵ)=r2 ℵ =r ℵ. Hence by the principle of mathematical induction rn ℵ =0 or rn ℵ =r ℵ for any r∈ℛ, r≠0, n∈Z+. Therefore, ℵ is semisecond submodule. Remarks and Examples (3):- (1) Every second submodule is semisecond. Proof: -Let ℵ be a S.R.M. ℬ such that ℵ is second submodule, that is r ℵ =0 or r ℵ = ℵ for every r∈ℛ, r≠0. If r ℵ =0, then r2 ℵ =r(r ℵ)=r(0)=0. If r ℵ = ℵ, then r2 ℵ =r(r ℵ)=r ℵ, that is r2 ℵ =0 or r2 ℵ =r ℵ so ℵ is semisecond by proposition (2.2). The converse of this remark is not true in general for example: - Consider the Z-module Z8, let ℵ =<2>, take r=2, r ℵ ={0,4}. Thus r ℵ ≠ ℵ and r ℵ ≠(0), that is ℵ is not second submodule, while for every r∈Z, r≠0, such that r is even, then r=2k for some k∈Z, so r2 ℵ =(2k)2 ℵ =0.Also if r is odd, then r=(2k+1), so r2 ℵ =(4k2+4k+1) ℵ = ℵ and r ℵ =(2k+1) ℵ =2k ℵ + ℵ = ℵ. Thus r2 ℵ =r ℵ. Thus ℵ is semisecond submodule. (2) The submodule Z of the Z-module Q is not semisecond submodule, but Q is a semisecond submodule of Q. (3) Any submodule of Zp∞ as Z-module is not semisecond submodule. (4) Let ℵ be a non-zero S.R.M. ℬ s.t. ℛ is a field, then ℵ is semisecond. Proof: - Let r∈ ℛ, r≠0 and suppose r2ℵ≠0. To prove r2ℵ=rℵ, let rn∈rℵ, then rn=r2(r-1n)∈r2ℵ, hence rℵ⊆r2ℵ, which implies that r2ℵ=rℵ. Thus ℵ is a semisecond submodule. (5) Let f: ℬ→ℬ' be an R-homorphism and ℵ is a semisecond submodule of ℬ, then f(ℵ) is a semisecond submodule of ℬ'. Proof: - Since ℵ is semisecond, then r2ℵ=rℵ or r2ℵ=0. Hence either f(r2ℵ)=f(rℵ) or f(r2ℵ)=f(0). Thus r2f(ℵ)=rf(ℵ) or r2f(ℵ)=f(0). Therefore, f(ℵ) is a semisecond submodule of ℬ'. (6) The inverse image of semisecond submodule need not to be a semisecond, for example: - Let Π:Z →Z/<6>≅Z6, <2> is semisecond submodule in Z6 but Π-1(2)=2Z is not a semisecond. The opposite of remark and example (2.3. (1)) is true under the class of torsion free module over an integral domain, where a module ℬ over an I.D. is called torsion free if τ(ℬ)=0, where τ(ℬ)= {m∈ ℬ; r∈ℛ, r≠0, rm=0}, see [2.P.45]. Proposition (4):-If ℵ is a semisecond S.R.M. ℬ such that ℬ is torsion free over an I.D. R, then ℵ is a second submodule. Proof:- let r∈ ℛ, r≠0. Since ℵ is semisecond submodule, then r2ℵ=0 or r2ℵ=rℵ. If r2ℵ=0, then r2=0 (since M is torsion free) and since ℛ is an I.D., then r=0, which is contradiction. Thus r2ℵ=rℵ and for any n∈ℵ, hence  �́�∈ℵ s.t. r2�́�=rn. Thus r(n-r�́�)=0. Since r≠0 and M is torsion free, then (n- �́� n)=0, that is n=r �́� , hence ℵ⊆rℵ and so, rℵ=ℵ. Therefore, ℵ is a second submodule. Corollary (5):- If ℬ is a torsion free over an integral domain, then ℵ is second submodule of ℬ if and only if ℵ is semisecond. Mathematics | 104 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.1999 Vol. 31 (3) 2018 Recall that a module ℬ is called multiplication if every submodule ℵ of ℬ,  an ideal I of ℛ s.t. I ℬ =ℵ, amounting to for every submodule ℵ of ℬ, ℵ=[ℵ : ℛ ℬ]. ℬ, see[3]. Proposition (6):- If ℬ is a faithful F.G. multiplication R-module, ℵ< ℬ, then ℵ is semisecond iff [ℵ:ℬ] is semisecond ideal of R. Proof:- (⟹) If ℵ is a semisecond submodule, then for any r∈ ℛ, r≠0, r2ℵ=rℵ or r2ℵ=0. If r2ℵ=rℵ, then r2[ℵ:ℬ]. ℬ =r[ℵ:ℬ]. ℬ because ℬ is a multiplication module. Since ℬ is a F.G. faithful multiplication ℛ -module, then by [1] r2[ℵ : ℬ]=r[ℵ:ℬ]. If r2ℵ=0, then r2[ℵ:ℬ]. ℬ =0 and hence r2[ℵ:ℬ]⊆ ℬ =0. Thus r2 [ℵ:ℬ]=0 and so [ℵ:ℬ] is a semisecond ideal. Now, to prove the opposite. Let [ℵ : ℛ ℬ] be a semisecond ideal, that is [ℵ : ℛ M] is a semisecond submodule of the ℛ -module ℛ. Then by proposition (2.2)  r∈R, r≠0, r2[ℵ : ℛ ℬ]= r[ℵ:ℬ] or r2[ℵ:ℬ]=0, that is r2[ℵ : ℛ ℬ] ℬ =r[ℵ : ℛ ℬ]. ℬ or r2[ℵ : ℛ ℬ]=0. Since ℬ is a multiplication module, we have r2ℵ=rℵ or r2ℵ=0 for every r∈ ℛ, r≠0. Therefore, ℵ is a semisecond submodule. We notice that the provision M is faithful cannot be dropped from proposition (2.6) for instance: Consider the Z-module Z6, Z6 is F.G. multiplication Z-module but not faithful. However, the submodule ℵ=<3> is a semisecond submodule since for any r2∉ ℛ ℵ=2Z, r2ℵ=rℵ. But [ℵ : ℛ ℬ]=[(3) : Z6] =3Z is not semisecond in Z, Since for every r2∉ ℛ (3Z)=0 and for each r ≠ ∓1 we have r2 (3Z) ≠ r(3Z). Proposition (7):- N.Z. ℵ S.R.M. ℬ is a semisecond ℛ -submodule iff ℵ is a semisecond ℛ /I-submodule, where I⊆ ℛ ℵ. Proof :- ⟹ Let �̅� =r+I ∈ℛ = ℛ /I. (�̅� )2ℵ=(r+I)2ℵ= rℵ . But r2ℵ=0 or r2ℵ=rℵ, since ℵ is semisecond, therefore (�̅� 2ℵ=0 or (�̅�)2ℵ= �̅�ℵ. Thus ℵ is a semisecond ℛ-submodule. Similarly, we can proof the opposite. Hence, we have the following result. Corollary (8):- If ℵ is a N.Z. S.R.M. ℬ is a semisecond submodule iff ℵ is a semisecond submodules ℛ / ℛ ℵ – submodule. Proposition (9):- Let ℵ be N.Z. proper submodule of ℬ s.t. ℛ ℵ is a maximal ideal, then ℵ is a semisecond submodule. Proof:- since ℛ ℵ is a maximal ideal, then ℛ/ ℛ ℵ is a field and by remark and example (2.3.(4)) ℵ is semisecond submodule ℛ / ℛ ℵ-submodule. Thus by corollary (2.8), ℵ is a semisecond submodule ℛ -submodule. Mathematics | 105 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.1999 Vol. 31 (3) 2018 Remark (10):- If ℵ=ℵ1⊕ℵ2 is semisecond submodule in ℬ=ℬ1⊕ℬ2, then N1 and N2 are semiseconds in ℬ 1, ℬ 2 respectively. Proof:- It follows directly by remark and example (2.3.(5)). Remark (11):- Let ℬ = ℬ 1⊕ ℬ 2. If ℵ1 and ℵ2 are semisecond submodules in ℬ 1 and ℬ 2 respectively, then it is not necessarily that ℵ 1⊕ ℵ 2 is semisecond submodule in M for example:- Let ℬ =Z6⊕Z16, let ℵ=<3>⊕<2>, <3> is semisecond submodule in Z6, <2> is semisecond submodule in Z16. However 2 ℵ =< 0 >+< 4 >, (22) ℵ =4 ℵ =< 0 >⊕< 8 >, then 22 ℵ ≠2 ℵ and 22ℵ≠<0>⊕<0>. The following result shows the direct sum of two semisecond submodules under certain condition. Proposition (12):- Let ℵ1 and ℵ2 be semisecond submodules in ℬ1 and ℬ2 respectively such that ℛ ℵ1 = ℛ ℵ2. Then ℵ1⊕ℵ2 is semisecond submodule in ℬ=ℬ1⊕ℬ2. Proof:- Let r∈ ℛ, r≠0, then (r2ℵ1=rℵ1 or r2ℵ1=0) and (r2ℵ2=rℵ2 or r2ℵ2=0). Suppose r2ℵ1=0, then r2ℵ2=0 since ℛ ℵ1 = ℛ ℵ2 and so r2(ℵ1⊕ℵ2) =0. If r2ℵ1=rℵ1 and r2ℵ1≠0, hence r2ℵ2≠0 so r2ℵ2=rℵ2. It follows that r2(ℵ1⊕ℵ2) = r2ℵ1⊕r2ℵ2=rℵ1⊕rℵ2. Now, we survey the relationships between semisecond submodules and some kind of submodules. A submodule ℵ of a module ℬ is rendering semiprime if ℵ≠ℬ and r∈ℛ, m∈ℬ, k∈Z+ with rKm∈ℵ, then rm∈ℵ,see[4].Equivalently ℵ is semiprime if whenever r∈ℛ, m∈ℬ, r2m∈ℵ, then rm∈ℵ,see[3,prop.(1.2)]. An R-module ℬ is rendering semiprime if (0) is a semiprime submodule of ℬ. Proposition (13):- Let ℬ be a semiprime ℛ-module, ℵ submodule of ℬ if ℵ is semisecond, then ℵ is semiprime submodule of ℬ. Proof:- Let a2x∈ℵ, where a∈ℛ, x∈ℬ. to prove ax∈ℵ. Since ℵ is semisecond, then either a2ℵ=0 or a2ℵ=aℵ. Assume a2ℵ=(0). Put a2x=n for some n∈ℵ. Then a4x=a2n∈a2ℵ=0, hence ax=0∈ℵ (since ℬ is semiprime). Assume a2ℵ=aℵ. Since a2x=n∈ℵ, then a3x=an∈aℵ=a2ℵ, so that a3x=a2n1for some n1∈ℵ. Hence a2(ax-n1)=0.As ℬ is semiprime a(ax1-n1)=0 and so that a2x=an ∈aℵ=a2ℵ. Thus a2ℵ=a2n2 for some n2∈ℵ. This implies a2(x-n2) =0. But ℬ is semiprime, so that a(x-n2) =0. It follows that ax=an2∈ℵ. Therefore, ℵ is a semiprime submodule. Note that the opposite of previous proposition is not hold in public for instance: - Take ℬ=Z as Z-module. ℬ is prime so it is semiprime. Let ℵ=<6> is semiprime, but N is not semisecond since for every r∈Z, r≠0, r2ℵ≠(0) and r2ℵ≠rℵ. Reminiscence that a module ℬ is rendering Coprime if ℛ ℬ = ℛ ℬ ℵ for every proper submodule ℵ of ℬ, see [5]. Equivalently ℬ is coprime module if and only if ℬ is second module, see [6, th.(2.1.6)]. Mathematics | 106 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.1999 Vol. 31 (3) 2018 A submodule ℵ of an ℛ-module ℬ is rendering irreducible if ℵ cannot be expressed as a finite intersection of proper divisors of ℵ, See [4]. Proposition (14):- Let ℬ be a coprime module, let N be a submodule of ℬ such that ℵ is irreducible. If ℵ is semiprime, then ℵ is second and hence semisecond. Proof:- Let ℵ be a semiprime ℛ-submodule, since ℵ is irreducible, then by [3,prop.(1-10)] ℵ is prime, but ℬ is coprime module, then by [6,prop(2.4.7)] ℵ is second, hence ℵ is semisecond. Corollary (15):- Let ℬ be a prime module over regular ring ℛ (in sense of von Neuman), let ℵ be a submodule of ℬ such that ℵ is irreducible.Then N is semisecond if and only if ℵ is semiprime. Proof:(⟹) Since ℬ is prime, so it is semiprime. Thus we have the result by proposition (2.13). (⟸) Since ℬ is prime module over regular ring, then by [6, corollary (2.4.3)] ℬ is coprime, hence we have the result by proposition (2.14). Reminiscence that a submodule ℵ of a module ℬ is rendering secondary (dual notion of primary module) if for each r∈R, the homothety r* on ℵ is either surjective or nilpotent, where r* is nilpotent if there exist k∈Z+, such that (r*)k=0, see[7].It is obvious that every second submodule is secondary, but the opposite is not whole in public. The next lemma explains that the opposite is whole under certain condition. Lemma (16):- Let ℵ be an ℛ-submodule such that ℛ ℵ is semiprime ideal. If ℵ is secondary, then ℵ is second submodule and hence semisecond. Proof:- Since ℵ is secondary, then for any r∈ℛ, r≠0, rℵ=ℵ or rnℵ=0; n∈Z+. If rℵ=ℵ, then there is nothing to prove. If rnℵ=0, then rn∈ ℛ ℵ. But ℛ ℵ is semiprime, so r∈ ℛ ℵ. Thus rℵ=0 and hence ℵ is second. Corollary (17): - Let ℵ be a S.R.M. ℬ such that ℛ ℵ is semiprime, then ℵ is secondary if and only if ℵ is second. The opposite of corollary (17) need not to be whole in public for example: - In Z8 as Z-module, <2> is Semisecond and not secondary. The opposite is whole under the class of torsion free module over regular ring. Remark (18): - If ℵ is semisecond submodules of torsion free module ℬ over regular ring, then ℵ is secondary. Proof:- The proof directly by proposition (4). Corollary (19) :- Let ℬ be torsion free over regular ring, let ℵ be submodule of ℬ such that ℛ ℵ is semiprime, then ℵ is secondary if and only if ℵ is semisecond. Now, we turn our attention to the localization of semisecond. Proposition (20):- Let ℵ be a semisecond submodule of an ℛ-module ℬ, then ℵs is semisecond ℛs-submodule of ℬs, s.t. S is a multiplicatively closed subset of R. Proof:- Let �̅�∈ℛs, �̅� = , where r∈ℛ, s∈S. Assume that (�̅�)2∉ ℵs .To prove (�̅� 2ℵs=�̅�ℵs. Since (�̅�)2∉ ℵs, then ( 2.( )≠ for some n∈ℵ, a∈S. ( )≠ , that is for any t∈s, r2tn≠0. Thus Mathematics | 107 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.1999 Vol. 31 (3) 2018 r2t∉ ℛ ℵ which implies that r2∉ ℛ ℵ, so r2ℵ≠0.But ℵ is semisecond, hence r2ℵ=rℵ. Therefore (r2ℵ)s =(rℵ)s. Thus (r2)s ℵs = (r)sℵs and so (�̅�)2ℵs=�̅�ℵs. Corollary (21):- Let ℵ be a semisecond submodule of an R-module ℬ, then ℵp is semisecond ℛp-submodule of ℬp for any prome ideal P of ℛ. 3. Semisecond Modules Yass in [1] introduced the notion of second module (where ℬ is second if for every r∈ℛ, r≠0, rℬ=0 or rℬ=ℬ). Equivalently ℬ is second module if ℬ is second submodule of ℬ. In this section we introduce the notion of semisecond module as a generalization of second module. We give some properties of semisecond module. Definition (22):- Let ℬ be an R-module, ℬ is rendering semisecond if ℬ is semisecond submodule, that is for any r∈ℛ, r≠0, r2ℬ=rℬ or r2M=0. Remarks and Examples (23) (1) It is obvious that every second module is semisecond, by remark and example (2.3.(1)).The opposite is not whole in public for instance: Z4 as Z-module is not second since 2Z4≠Z4 and 2Z4≠(0) but Z4 is semisecond module. (2) Z as Z-module is not semisecond , since for any r∈ℛ, r≠0, r2Z≠(0) and r2Z≠rZ. (3) Consider the Z-module Zp∞, Zp∞ =0, that is for all r∈Z, r≠0, r2Zp∞≠(0). But Zp∞ is divisible Z-module, so r2Zp∞=rZp∞; for all r∈Z, r≠0, then Zp∞ is semisecond. (4) Q as Z-module is semisecond module. (5) If n is a prime number, then Zn is semisecond Z-module, but the opposite is not whole in public for example Z6 is semisecond but 6 is not prime. (6) A module ℬ is semisecond ℛ-module iff ℬ is semisecond ℛ/I-module, where I⊆ ℛ ℬ. Proof :- It follows by proposition (7). (7) A module ℬ is semisecond ℛ-module iff ℬ is semisecond ℛ/ ℛ ℬ-module. Proof :- It follows by corollary (8). (8)Let f:ℬ⟶ ℬ ' be an R-homomorphism, if ℬ is semisecond module, then f(ℬ) is semisecond ℬ'-module. (9) Let ℬ be a semisecond ℛ-module, then ℬs is semisecond ℛs-module, s.t. S is a multiplicatively closed subset of ℛ. Proof :- It holds by proposition (20). (10) Let ℬ be a semisecond ℛ-module, then ℬp is a semisecond ℛp-module for any prime ideal P of ℛ. Proof:- It follows by corollary (21). Mathematics | 108 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.1999 Vol. 31 (3) 2018 References 1. Yassemi, S. The Dual Notion of Prime Submodules. Arch. Math. (Born). 2001, 73, 273-278. 2. Abdul-Baste, Z.; Smith, P.F. Multiplication Modules. Comm. In Algebra. 1988, 16, 755-779. 3. Athab, I.A. Prime Submodules and Semiprime Submodules. M.Sc. Thesis, University of Baghdad. 1996. 4. Dauns, J. Prime Modules and One Sided Ideals in Ring Theory and Algebra III. Proceedings of the third oklahomo conference. 1980, 301-344. 5. Annin, S. Associated and Attached Primes over Non Commutative Rings. Ph. D Thesis, University of Berkeley. 2002. 6. Rasha, I.k. Dual Notions of Prime Submodules and Prime Modules. M.Sc. Thesis. University of Baghdad. 2009. 7. MacDonald, L.G. Secondary Representation of Modules over Commutattive Ring. Sympos. Math. XI, 1973, 33-43.