IHJPAS. 36 (4) 2023 377 This work is licensed under a Creative Commons Attribution 4.0 International License Abstract Suppose that A is an abelain ring with identity and B is a unitary (left) A-module. In this paper, we introduce a type of module, namely quasi-semiprime. A-module, whenever √[𝑁: 𝐡] is a prime ideal for proper submodule N of B, then B is called quasi -semiprime module, which is a generalization of quasi-Prime A-module, whenever annAN is a prime ideal for proper submodule N of B, then B is quasi-prime module. A comprehensive study of these modules is given, and we study the relationship between quasi-semiprime modules and quasi-prime. We put the condition coprime over cosemiprime ring for the two cocept quasi-prime modules and quasi-semiprime modules, which are equivalent. The concepts of prime modules and quasi-semiprime modules are equivalent. The condition of anti-hopfain makes quasi-prime is quasi-semiprime A-module. Whenever B is cyclic, coprime C-module, where C is the ring, each ideal is semiprime, which implies quasi-prime, quasi-simepime, and annCB are prime ideals. If F is an epimorphism from B1 β†’ B2, whenever B1 is a quasi-prime module, it implies B2 is a quasi-prime A-module, and the inverse image of quasi-semiprime is a quasi-prime A-module. Keywords: Prime module, Quasi-prime R-modules, Quasi-semiprime R-modules, Coprime R- modules, Antihopfian R-modules. 1. Introduction Suppose that W is a left A-module, where A is a ring with unity. An A-module B is said to be prime whenever annAB=annAN for each non-zero submodule Nof B, where annAB={a ∈ A;bx=0 for each b ∈ B}[1,2]. Hasan in [3] introduced the concept of qasi-prime A-modules, which is a generalization of prime A-modules, where an A-module W is called quasi-prime modulles if and only if for each non-zero submodule N of W, annA N is a prime ideal. Annin [9] calls an A-module W a coprime (dual notion of prime modules) if annAW=annAW/A for every proper submodule A of W. In this paper, we study a generalization of the quasi-prime module which we called the Quasi-semiprime A-module if √annB/N = √[N: B] is a prime ideal for each submodule N of B. This paper consists of two sections. In section one; we study the basic properties of a quasi- doi.org/10.30526/36.4.3181 Article history: Received 9 January 2023, Accepted 20 February 2023, Published in October 2023 Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq Quasi-Semiprime Modules Muntaha Abdul- Razaq Hasan Department of Mathematics, College of Basic Education, Mustansiriyah University, Baghhdad-Iraq https://creativecommons.org/licenses/by/4.0/ mailto:muntha_1974rzaq.edbs@uomustansiriyah.edu.iq IHJPAS. 36 (4) 2023 378 semiprime A-module. In section two, we study the relation between quasi-semiprime A- modules and prime A-modules. 2. Materials and Methods Definition (2.1) B is said to be a quasi-semiprime A-module if √[N: W] is a prime ideal for the proper submodule N of B. Examples and Remarks (2.2) 1- It is clear that Zn is a quasi-semiprime A-module if and only if n is a prime number. 2- If n can be written as a product of two prime numbers, then Zn is a quasi-semiprime A-module. Proof: Let n=p1p2; p1, p2 be two prime numbers, so N1= (p1),N2=(p2), then√[(𝑃1: 𝑍𝑛] =√(𝑝1) = (p1), N2=(p2), then √[(𝑝2):𝑍𝑛] = √(𝑝2) = (p2) is a prime ideal, henc Zn is a quasi-semiprime A- module. 3- Zp∞ is not a quasi-semiprime module, since we know that every submodule of Zp∞ is of the form (1/pn +Z), where n is a non-negative integer, so √[ 1 𝑝𝑛 + 𝑍: π‘π‘βˆž] = √[ 1 𝑃𝑛 + 𝑍] = (pn Z) is not prime ideal. 4- Suppose B is a simple A-module, then B is a Quasi-Semiprime A-Module. Proof: it is clear. Proposition (2.3) Every proper submodule N of quasi-semiprime module is a quasi-semiprime module. Proof: Suppose N is a proper submodule of quasi-semiprime A-module W. Let K be a proper submodule of N to show that √[𝐾: 𝑁] is a prime ideal if ab∈ √[𝐾: 𝑁], so anbn∈ [K:N], so that anbnN βŠ† K βŠ† W that is anbn ∈ [N:W], but W is a quasi-semiprime A-modul implies either an ∈ [N:W] or bn ∈ [N:W], thus either an∈ [K:N] or bn ∈ [K:N] which means either a∈ √[𝐾: 𝑁] or b∈ √[𝐾: 𝑁], so √[𝐾: 𝑁] is a prime ideal. Recall that whenever B β‰… B/N for all proper submodule Nof modules B, then we said that anon- simple A-module B anti-hopfian module [4, 5]. Proposition (2.4) Suppose that B is an anti-hopfian quasi-prime A-module, then B is quasi-semiprime. Proof: Since B is an anti-hopfian module, then B β‰… B/N for N be a proper submodule of B, so there exists an isomorphism function f: Bβ†’ B/N; f(b)=b+N for each b ∈ B, so it is easy to check that IHJPAS. 36 (4) 2023 379 annAB=annAB/N, then by [6], every anti-hopfian A-module is a coprime E-module, where E=End(W) and by [5] every f ∈ E, either f=0 or f is subjective, thus f(b)=0 or f(b)=B for every w ∈ W If f(w)=0 implies W=N which is a contradiction, so f(W)=W, which means annAB=[N:B], implies βˆšπ‘Žπ‘›π‘›π΄π΅ =βˆšπ‘Žπ‘›π‘›π‘ 𝐡 if anbn ∈[N:B], then anbn ∈ annAB but B is a quasi-prime A-module, so by [3] implies annAW is a prime ideal, so either an∈ annAW or bn ∈ annAB. Thus, either a ∈ √[𝑁: 𝐡] or b∈ √[𝑁: 𝐡], which means B is a quasi-semiprime A-module. The condition anti-hopfian we cannot drop for example: Z6 is quasi-semiprime A-module by (2.2), while it is not quasi-prime by [3], and Z6 is not anti-hopfain by [6]. Prorosition (2.5) Suppose B is a coprime A-module of quasi-prime, then B is quasi-semiprime A-module. Proof Let B be a quasi-prime A-module, then by [3], annAB is a prime ideal, but W is a coprime A- module, so annAB/N is a prime ideal for each non-zero submodule N of B, which means√[𝑁: 𝐡] is a prime ideal. Thus, W is a quasi-semiprime A-module. Recall that an ideal K of the ring A is called nil radical and denoted by√𝐾, and is defined by: √𝐾 ={a∈ A; an∈ K , for some n Z+}[8]. Not (2.6) Suppose C is a ring where every ideal is nil radical, which we call cosemiprime ring. Theorem M (2.7) Suppose that B is a coprime C-module. The following statements are equivalent: 1) B is a Quasi-Prime Module. 2) B is a Quasi-Semiprime Module. Proof: 1) β†’ (2) (by Theorem ( 2-5)) (2) β†’ ( 1) for each a,b ∈ C if ab annC N,then abN=0 implies ab ∈ [(0):N],which means ab∈ √[(0): 𝑁], but W is quasi-semiprime module so either a∈ √[(0): 𝑁] or b∈ √[(0): 𝑁] implies either a∈ annCN or b∈ annCN. Thus, B is a quasi-prime C-module. Theorem (2.8) Let B be a cyclic coprime C-module, then the following statements are equivalent: 1- B is a Quasi-Prime C-Module. 2- B is a quasi-semiprime C-module. 3- annCB is a prime ideal. IHJPAS. 36 (4) 2023 380 Proof: 1β†’ 2 (by Theorem (2.5)) 2 β†’ 3 if ab ∈ annCB, then ab∈ βˆšπ‘Žπ‘›π‘›π΅. Thus, ab∈ √[𝑁: 𝐡] for every submodule N of B which means anbn ∈ [N:B], but B is a quasi-semiprime C-module, so either an∈ [N:B] or bn ∈ [N:B], but B is coprime by [9] implies either anB=0 or bnB=0, which means either a∈ βˆšπ‘Žπ‘›π‘›π΅ or b∈ βˆšπ‘Žπ‘›π‘›π΅.Thus, either a∈ annCW or b∈ anncB. 3 β†’ 1 by [3] implies the result. Proposition (2.9) Suppose that B is an A-Module and J is an Ideal Of A which that is contained in annAB/N where N is a submodule of B. Then, B is a quasi-semiprime A-module ⟷ B is a quasi-semiprime A/J- Module. Proof To show B is quasi-semiprime A\J-module if (a1+J)(a2+J)∈ √[𝑁:𝐴/𝐽 𝐡], where a1+J,a2+J∈ B/J, then (a1a2+J)n∈ [N:A/IB]. Thus, (a1 na2 n+J)x=0 for all x∈ annA/JB/N. Hence, a1 na2 nx=0 for all x∈ annAB/N which means a1 na2 n∈ [N:B], so a1a2 ∈ √[𝑁:𝐴 𝐡], but B is quasi-semiprime A-module, which implies either a1 ∈ √[𝑁:𝐴 𝐡] or a2∈ √[𝑁:𝐴 𝐡]. Thus, either a1 n ∈ [N:𝐴B] or a2 n ∈[N:𝐴B]. However, a1 n+I ∈ annAB/N or a2 n+J ∈ ann B/N. Thus, either (a1+I)∈ √[𝑁:𝐴/𝐽 𝐡] or (a2+I)∈ √[𝑁:𝐴/𝐽 𝐡], which means B is a quasi-semiprime B/I module. Conversely ,if B is quasi-semiprime A/J-module ,let N be a nonzero A-submodule of B,let a1,a2∈ √[𝑁:𝐴 𝐡] ,then a1 n a2 nx=0 for all x ∈annAB/N.Hence (a1 n+J)(a2 n+J)x=0 for all x∈ annA/JW/N ,so(a1+J)(a2+J)∈ √[𝑁:𝐴/𝐽 𝐡] ,whille is a prime ideal,so either (a1+I)∈ √[𝑁:𝐴/𝐽 𝐡] or (a2+I)∈ √[𝑁:𝐴/𝐽 𝐡] .Then we get either a1 n x=0 or a2 nx=0 f0r each x ∈ annAB/N,so either a1∈ √[𝑁:𝐴 𝐡] or a2 ∈ √[𝑁:𝐴 𝐡] . Theorem (2.10) Suppose B1, and B2 are two A-modules, if f: B1 β†’ B2, is an epimorphism function, then if B1 is a quasi-semiprime module, then B2 is a quasi-semiprime A-module. Proof: Since B1 is a quasi-semiprime A-module, so if anbnB1βŠ† N1 for each a,b∈ A, then either anB1βŠ† N or bnB1βŠ† N. Thus, f(anbnB1)βŠ† f(N1) since f is a homomorphism implies f(an).f(bn)∈ [f(N):f(W1)]. Suppose f(a)=x,f(b)=y. Thus, either f(anB1) βŠ†f(N1) or f(bnB1)βŠ† f(N), so either f(an)f(W1) βŠ†f(N) or f(bn)f(B1)βŠ† f(N) implies either xnf(B1)βŠ† f(N) or ynf(B1)βŠ† f(N1), but f is onto, so f(B1)=B2,f(N)=N2, which means either x∈ √[𝑁2∢𝐡2 or y∈ βˆšπ‘2: 𝐡2, whenever xy∈ βˆšπ‘2: 𝐡2 . Thus, βˆšπ‘2: 𝐡2 is a prime ideal, which means B2 is a quasi-semiprime module. IHJPAS. 36 (4) 2023 381 Corollary (2.11) The inverse image of the quasi-semiprime module is a quasi –semiprime module. Theorem (2.12) Let B1 and B2 be two quasi-semiprime A-modules such that for each proper submodule K,T of B1,B2, respectively, if [K⨁ T:W]=[K:W]∩ [T:W], then B=B1⨁ B2 is a quasi-semiprime A- module, where√[𝐾: 𝐡] βŠ† √[𝑇: 𝐡] or √[𝑇: 𝐡] βŠ† √[𝐾: 𝐡] . Proof We must prove √[𝐾⨁𝑇: 𝐡] is a prime ideal for the proper submodules K, T of B1 and B2 in the order. Since √[𝐾 ⨁𝑇: 𝐡] =√[𝐾: 𝐡] ∩ √[𝑇: 𝐡] where either√[𝐾: 𝐡] βŠ† √[𝑇: 𝐡]. Or√[𝑇: 𝐡] βŠ† √[𝐾: 𝐡]. Thus, either √[𝐾⨁𝑇: 𝐡] =√[𝐾: 𝐡] or √[𝐾⨁𝑇: 𝐡] =√[𝑇: 𝐡], but W1, and W2 are quasi-semi-prime modules. Therefore,√[𝐾: π‘Š], and √[𝑇: π‘Š] are prime ideals in A. Implies √[𝐾⨁𝑇: 𝐡] is a prime ideal in A. Thus, B1⨁ B2 is quasi-semiprime A-modules. The condition √[𝐾: 𝐡] βŠ† βˆšπ‘‡: 𝐡] or √[𝑇: 𝐡] βŠ† √[𝐾: 𝐡] we cannot be dropped, for example, let B1=Z6, and B2=Z3 are two quasi-semiprime A-modules by (Examples and Remark (2.2), √[(2): 𝑍18] ⊈ √[(3): 𝑍18 and √[(3): 𝑍18 ⊈ √[(2): 𝑍18 Since√(3) ⊈ √(2) and √(2) ⊈ √(3) so√[𝑍2⨁ 𝑍3: 𝑍18 β‰  βˆšπ‘2: 𝑍18 ∩ βˆšπ‘3: 𝑍18 =√9𝑍 ∩ √6𝑍 =(3) ∩(6)=(6) is not a prime ideal, implying W =W1 ⨁W2 is not a quasi-semiprime module. 3. Quasi-Semi-Prime A-Module and Prime Module Now, we turn our attention to the relationship between quasi-semiprime modules and prime modules. Proposition (3.1) Suppose B is a coprime A-module, then every prime A-module is a quasi-semiprime A-module. Proof It follows directly by from [3] and Propositions (2.5). The next example shows that the converse of Proposition (3.1) is not valid in general. Let Z6 as a Z –module is quasi-semiprime module by Examples and Remarks (2.2), while it is not a prime module [1]. IHJPAS. 36 (4) 2023 382 Theorem (3.2) Suppose B is a coprime C-module, then the following statements are equivalent: 1-B is a prime C-module. 2-B is a quasi-semiprime C-module. Proof: 1β†’ 2 by (Proposition (3-1)), 2 β†’ 1 because B is a quasi-semiprime C-module, so √[𝑁: 𝐡] is a prime ideal for each N submodule of M, so [N:B] is a prime ideal, but B is a coprime C-module, so [6] implies annCB is a prime ideal, which means if rb=0 for b∈ B and c∈ C. Suppose that bβ‰  0 and Cbβ‰  o, so cB=Nβ‰  0, thus there exists that b∈ B and n∈ N such that cb=n, this means N=0, which is a contradiction. So B is a Prime. Proposition (3.3) Let B be a coprime C-module, then the following statements are equivalent: 1- Bis a quasi-prime module. 2- B is a quasi-semiprime modul. 3- B is a prime module. Proof 1 β†’ 2 by Theorem(2.7). 2 β†’ 3 by Theorem (3.2). 3β†’ 1 by [3]. Corollary (3.4) If B is a coprime C-module, then B is a quasi-semiprime C-Module ⟷ (0) is a prime C- submodule. Proof It is clear. Conclusion From this research, we introduced a new definition of quasi-semiprime modules and studied the relationship between quasi-semiprime modules and other modules, such as quasi-prime modules and prime modules. If we put the condition coprime, the cocept quasi-prime module, quasi- semiprime module, and prime module are equivalent. References 1. AL-Bahraany, B. Anote on Prime Modules and Pure Submodules, J. sclence 1996, 37, 2, 1431-1441. 2. Desale, G.;, Nicholson,W. K., Endoprimitive Ring , J. Algebra 1981,70,3,548-560. IHJPAS. 36 (4) 2023 383 3. Hasan,M.A.Quasi-prime module and Quasi prime submodule,M.SC..Thesis 1999, Univ.of Babhdad. 4. Hirano,Y. ;Mogani,I.On Restricted Anti-Hopfinan Modules,Math.J., Kayama1986 ,Univ.,.28,119-131. 5. AL-Awadi, H.K.Anti-Hopfian Modules and Restricted Anti-Hopfian, M.SC. thesis 200,Univ. of Baghdad. 6. Hadi.M. A.I,; Kassm, I. R.,Coprime Modules And Other Related Topics,, Journal of physics 2018 1003,1,1-15. 7. Hadi M. A. I ;Kasam,.I.R. 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