ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Hai tham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 On Weakly Quasi-Prime Module M. A. Hassin Departme nt of Mathematical, College of Basic Education, Unive rsity of Al-Mustansriyah Received in: 3 April 2011 Accepte d in: 18 October 2011 Abstract In this work we shall introduce the concept of weakly quasi-prime modules and give some prop erties of this ty p e of modules. Key words: Prime module, quasi-p rime module, weakly quasi-prime module. 1- Introduction Let R be a commutative ring with unity , and let M be an R-module, we introduce that an R-module M is called weakly quasi–prime module if annRM = annRrM for every r annRM , where annRM = {r: rR and rM = 0}. The main p urp ose of this work is to invest igate the p rop erties of weakly quasi-prime modules, and we give several characterizations of weakly quasi-prime modules. Recall that an R-module is called p rime if annRM = annRN for every non-zero submodule N of M and annRM = {r: rR and rM = 0}, [1]. A submodule N of M is said to be p rime if a m  N for a  R, m  M , then either m  N or a  [N:M ] where [N:M ] = {r: rR, rM  N}, [1], [2]. It was shown t hat in [1] M is p rime module iff (0) is p rime submodule. The concept of quasi-prime module is introduced in [3] where an R-module M is quasi-prime module if annRN is p rime ideal for every nonzero submodule N of M . If M is quasi-prime module then annRM = annRrM  r  annRM , [3]. But the converse is not true for examp le: Let M =  p Z as Z-module is not quasi-prime module since if N = <1/p 2 + z >   p Z . So annRN = p 2 z is not p rime ideal in Z. But ann  p Z = 0 and  r  0, let a  ann r  p Z so a r p Z  = 0, so a r  ann  p Z . a r = 0, but r  0 so a = 0 so ann r  p Z = 0. Then ann  p Z = ann r  p Z . 2- Weakly Quasi-Prime Module In this section we introduce the concept of weakly quasi-prime module and give several results about it. 2.1 Definition: An R-module M is called weakly quasi-prime module (briefly W.q.p) if annRM = annRrM for every r ann RM . Recall that if R is an integral domain, an R-module M is said to be divisible iff rM = M for every nonzero element r in R, [4,p.35]. 2.2 Examp les and Remarks: ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Hai tham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 1. If M is divisible over integral domain then M is W.q.p. 2. Every quasi-prime is W.q.p but the converse is not true (see the examp le in the introduction). 3. Z as Z-module is W.q.p module since annRZ = 0 = annR rZ,  r annRZ. 4. Z4 as Z-module is not W.q.p module Since annRZ4 = 4Z and annR2Z = annR( 2 ) = 2Z . Thus Z4 as Z-module is not W.q.p module. 5. Z6 as Z-module is not W.q.p module since annZ6 = 6Z and ann2Z6 = ann( 2 ) = 3Z, so annZ6 ann 2Z6. 6. Zn as Z-module is W.q.p module iff n is p rime. 7. Let M = ZZp; p is p rime number is W.q.p module since annM = ann rM = 0 for each r  ann(Z Zp). 8.  p Z is W.q.p module since ann  p Z = ann r  p Z = 0. 2.3 Note : Let M be W.q.p over integral domain in R. Then every divisible submodule of W.q.p module. Recall that a p rop er submodule N of M is called semi-p rime submodule if every r  R, x  M , k  Z+, such that r k x  N, t hen rx  N, [4,p .50]. 2.4 Proposi tion: Let M be divisible and (0) submodule of M is semi-p rime submodule, then the following st atements are equivalent 1. M is p rime module, 2. M is q.p module, 3. M is W.q.p module. Proof :(1) → (2), by [2,p 10] (2) → (3), by [2,p 20] (3) → (1) To p rove M is p rime module, i.e. to show that (0) is p rime submodule. Let rm = 0, r  R, m  M , to p rove either m = 0 or r  annRM . Sup p ose r  annRM , so we must p rove that m = 0. Since r  annRM , rM  0. Hence rM = M , because M is divisible. Thus m = rm1 for some m1  M . Since rm = r(rm1) = 0, that is r 2 m1 = 0 which imp lies that rm1 = 0, since (0) submodule of M is semi-p rime. Thus m = 0. 2.5 Remark: The condition in prop osition 2.4 is necessary as the following examp le shows:  p Z is not q.p since if N = 2 1 p + Z then ann N = p 2 Z is not p rime ideal, but  p Z is W.q.p module (see the examp le in the introduction). 2.6 Theorem: Let M be a module over an integral domain R and every submodule of M is divisible then ann (rm) =ann (m), for each r  ann (m). Proof: Since (rm)  (m), so ann(m)  ann (rm) …(1) To p rove ann (rm)  ann (m) Let x  ann (rm) so x (rm) = 0. Since every submodule of M is divisible, (rm) = (m) and so xm = 0 which imp lies x  ann (m). Thus ann(rm)  ann(m) …(2) From (1) and (2), we have ann (m) = ann(rm), for each r  ann (m). Recall that an R-module M is called multip lication R-module if for every submodule N of M , there exists an ideal I of R such that IM = N. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Hai tham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 2.7 Theorem: Let M be multip lication W.q.p R-module. Then every submodule of M is W,.q.p module. Proof: Let N be submodule of M , since M is multiplication R-module, so N = IM ; I be ideal of ring R. To p rove N is W.q.p module. To p rove annRN = annRrN,  r annRN since rN  N so annRN  annRrN …(1) To p rove annRrN  annRN. Let x  annRrN so xrN = 0. Since M is multiplication so there exists an ideal I of R such that N = IM . Thus xrIM = 0; that is xI  annRrM = annRM , hence xIM = 0; so xN = 0 which implies x  annRN. T hus annRrN  annRN …(2) From (1) and (2) we have annRN = annRrN so N is W.q.p module. 2.8 Prop osition: Let M be cy clic W.q.p R-module. Then M is q.p module. Proof: Let M be cy clic so t here exist x  M ; M = (x), let y  M , to p rove annRy is p rime ideal, so y = rx; r  R, let a, b  annRy, to p rove either a  annRy or b  annRy. Since ab  annRy = annRrx, so abrx = 0. Sup p ose b  annRy = annRrx, i.e brx ≠ 0, so ab  annR(rx) = annR(x), since M is W.q.p module, so abx = 0 which imp lies that a annRbx= annR(x) (since M is W.q.p). Thus ax = 0 which imp lies rax = r.0 = 0 so a  ann (rx) which means a  annRy. 2.9 Theorem: Let M be cy clic R-module then the following st atements are equivalent 1. M is p rime module 2. annRM = annRIM ; I ⊈ annRN 3. M is W.q.p module. Proof: To p rove (1) → (2) It is clear by definition of p rime submodules. (2)  (3) it is obvious. To p rove (3)  (1), to p rove M is p rime module. By p rop osition (2.8) we have M is q.p module which imp lies that annRM is p rime ideal, see [3,p .14] and by [3,p .8] we get M is a prime module. 2.10 Theorem: The direct sum of two W.q.p R-module is also W.q.p R-module. Proof: Let M = M 1  M 2 where M 1 and M 2 are two W.q.p module, to p rove M is W.q.p module, i.e to p rove annRM = annRrM , for all r  annRM . annRrM = annRr(M 1  M 2) = annR(rM 1  rM 2) , see [2, p.80] = annRrM 1  annRrM 2 , see [2, p.83] = annRM 1  annRM 2 , since M 1 and M 2 are W.q.p = annR(M 1  M 2) = annRM 2.11 Corollary: Let M be an R-module if M is W.q.p module then for any p ositive integer n, M n is W.q.p module where M n is the direct sum of n copies of M . 2.12 Remark: ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Hai tham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 A direct summand of W.q.p module is need not be W.q.p module. For examp le: Let M = Z  Z4 so annRM = annRrM  r  annRM . But Z4 is not W.q.p module, (see remarks and examp les (2.2(4)). 2.13 Theorem: Let M 1 ; M 2 then M 1 is W.q.p iff M 2 is W.q.p. Proof:  Let f: M 1 → M 2 be 1-1 and onto and homomorp hisim and M 2 is W.q.p. To p rove M 1 = f – 1 (M 2) is W.q.p module, that is to p rove annRrf – 1 (M 2)  annRf – 1 (M 2); rannRf –1 (M2), let x  annR r f – 1 (M 2) so xrf – 1 (M 2) = 0 and since f – 1 is homomorp hisim so f – 1 (xrM 2) = f -1 (0) and since f – 1 is 1-1 so xrM 2 = 0 which mean x  annRrM 2 but M 2 is W.q.p module and rannRM 2 then xM 2 = 0 which imp lies f – 1 (xM 2) = f – 1 (0), but f – 1 is homomorp hisim so x f – 1 (M 2) = 0 implies x  annRf – 1 (M 2) so annRr f – 1 (M 2)  annR f – 1 (M 2) …(1) and since r f – 1 (M 2)  f – 1 (M 2 ), so annRf – 1 (M 2)  annRr f – 1 (M 2) …(2) From (1) and (2) we have annRf – 1 (M 2) = annRrf – 1 (M 2). So f – 1 (M 2) is W.q.p module.  clearly. 2.14 Not e: The condition "isomorphism" in theorem 2.13 is necessary as the following examp le shows Examp le: Let : Z  Z ⁄ (4) ; Z4, where Z is W.q.p , but Z4 is not W.q.p. It is known that, if M is an R-module and I is an ideal of R which is contained in annRM then M is R/I-module, by taking (r + 1)x = rx x  M , r  R, see [5,p.40]. Now, we give the following result. 2.15 Theorem: Let M be an R-module and let I be an ideal of R, which is contained in annRM . Then M is W.q.p R-module iff M is W.q.p R/I-module. Proof:  To p rove M is W.q.p R/I-module, i.e. to p rove annR/IM = annR/I(r + 1)M . Since (r + 1)M  M so annR/IM  annR/I(r + 1)M …(1) To p rove annR/I(r + 1)M  annR/IM Let x  annR/I(r + 1)M so x(r + 1)M = 0, which imp lies (xr + 1)M = 0 so (xr)M = 0 (by definition), so x  annRrM = annRM (since M is W.q.p R-module). x  annR/IM (since I  annR/IM ), so annR/I(r + 1)M  annR/IM …(2) From (1) and (2) we have annR/IM = annR/I(r + 1)M .  If M is W.q.p R/I-module then M is W.q.p R-module, i.e. to p rove annRM = annRrM ,  r  annRM . Since rM  M so annRM  annRrM …(1) To p rove annRrM  annRM Let x  annRrM so (xr)M = 0 imp lies that (xr + 1)M = 0, so x(r + 1)M = 0, hence x  annR/I(r + 1)M = annR/IM (since M is W.q.p R/I-module). Thus x  annR/IM , which imp lies that x  annRM (since I  annRM ), so annRrM  annRM …(2) ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Hai tham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 From (1) and (2) we have annRM = annRrM . So M is W.q.p module. Recall that a subset S of a ring R is called multiplicatively closed if 1  S and ab  S for every a, b  S. We know that every p rop er ideal P in R is p rime if and only if R-P is multiplicatively closed, see [4,p.42]. Let M be a module on the ring R and S be a multiplicatively closed on R such that S  0 and let RS be the set of all fractional r/s where r  R and s  S and M S be the set of all fractional x/s where x  M , s  S; x1/s1 = x2/s2 if and only if there exists t  S such that t(s1x2 – s2x1) = 0. So, can make M S into RS-module by setting x/s + y/t = (tx + sy )/st, r/tx/s = rx/ts for every x, y  M and for every r  R, s, t  S. If S = R-P where P is a prime ideal we use M P instead of M S and RP instead of RS. A ring in which there is only one maximal ideal is called local ring, see [4,p .50], hence RP is often called the localization of R, similar M P is the localization of M at P. So we can define the two map s :R  RS, such that (r) = r /1, rR, :M  M S, such that (m) = m /1, mM , see [5,p .69]. Through this p aper S – 1 R and S – 1 M represent RS and M S resp ectively. 2.16 Prop osition: Let M be W.q.p R-module then S – 1 M is W.q.p S – 1 R-module for each multiplicatively closed set S of R. Proof: To p rove 1 S Rann  S – 1 M = 1 S Rann  r/t S – 1 M  r t  1 S Rann  S – 1 M , since r/t S – 1 M  S – 1 M so 1 S Rann  S – 1 M  1 S Rann  r/t S – 1 M …(1) To p rove 1 S Rann  r/t S – 1 M  1 S Rann  S – 1 M Let y/t '  1 S Rann  r/t S – 1 M so y/t 'r/t S – 1 M = 0 which imp lies that yr/tt 'S – 1 M = 0 where yr  M , tt '  S so yr/ tt 'S – 1 M = 0 which imp lies that yr/ tt 'M /S = 0 so yrM = 0. Hence y  annRrM = annRM . Since y  annRM so yM = 0. Thus yM /ts = 0 so y/tS – 1 M = 0, y/t annRS – 1 M , hence 1 S Rann  r/t S – 1 M  1 S Rann  S – 1 M …(2) From (1) and (2) we have 1 S Rann  S – 1 M = 1 S Rann  r/t S – 1 M , so S – 1 M is W.q.p module. Re ferences: 1. AL-Bahraany, B., (1996), Not e on Prime M odules and Pure Submodule, J.Science, 37, . 1431 – 1441. 2. Anderson, F.W. and Fuller,R.R., (1973), Rings and Categories of M module, University of Oregon. 3. Abdul Razak, H.M ., (1999), Quasi Prime M odule and Quasi-Prime Submodule. M .Sc. thesis, University of Baghdad. 4. Sharpe, D.W. and Vamos, P., (1972), Injective M odules, Cambridge University , p ress. 5. Larsen, M .D. and M ccarl, P.J., (1971), M ultip licative Theory of Ideals, Academic Press, New York. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Hai tham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 حول المودیوالت الشبه األولیه الضعیفة منتهى عبد الرزاق حسن قسم الریاضیات، كلیة التربیة االساسیة، الجامعة المستنصریة 2011 تشرین االول 18: قبل البحث في2011 نیسان 3: استلم البحث في الخالصة ُوقـد برهنـت بعـض الخـواص لهـذا النـوع مـن . ُ قدمت تعریف جدید وهو المودیوالت الشبه أولیه الـضعیفة في هذا العمل .المودیوالت . المودیول األولي ، المودیول الشبه األولي ، المودیول الشبه األولي الضعیف:الكلمات المفتاحیة