179 | Mathematics 2015) عام 1(العدد 28المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol. 28 (1) 2015 On Semi-Essential Submodules Muna A. Ahmed math.200600986@yahoo.com Maysaa R. Abbas maysaa.alsaher@yahoo.com  Department of Mathematics, college of Science for Women,University of Baghdad, Iraq-Baghdad. Received in : 20 October 2014 , Accepted in :5 January 2015 Abstract Let R be a commutative ring with identity and let M be a unitary left R-module. The purpose of this paper is to investigate some new results (up to our knowledge) on the concept of semi-essential submodules which introduced by Ali S. Mijbass and Nada K. Abdullah, and we make simple changes to the definition relate with the zero submodule, so we say that a submodule N of an R-module M is called semi-essential, if whenever N ∩ P = (0), then P = (0) for each prime submodule P of M. Various properties of semi-essential submodules are considered. Keywords: Essential submodules, Semi-essential submodules, Uniform modules, Semi- uniform modules, Fully prime modules and Fully essential modules. his paper is a part of a thesis submitted by the second author and supervised by the first 180 | Mathematics 2015) عام 1(العدد 28المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol. 28 (1) 2015 1. Introduction Throughout this paper, R represents a commutative ring with identity and M is a unitary left R-module. Assume that all R-modules under study contain prime submodules. It is well known that a submodule N of M is called essential, if whenever N ∩ L = (0), then L = (0) for each submodule L of M [7] and [9]. Ali and Nada in [1] introduced the concept of semi-essential submodules as a generalization of the class of essential submodules, where they say that a nonzero submodule N of M is called semi-essential, if N ∩ P ≠ (0) for each nonzero prime R- submodule P of M [1], where a submodule P of M is called prime, if whenever rm  P for r R and m M, then either m  P or r  (P: M) [12]. In this paper we rewrite the definition of the semi-essential submodules which introduced [1] in another formula, in fact we didn't find any reasonable reason to exclude the zero submodule from the definition of semi-essential submodules. Also we give some new results (up to our knowledge) about this concept, and illustrate that by some remarks and examples. We start by the formula of the definition of the semi-essential submodules. Definition (1.1): A submodule N of an R-module M is called semi-essential if whenever N ∩ P = (0), then P = (0) for every prime submodule P of M. We see it is necessary to put some simple remarks about the class of semi-essential submodules which not mentioned in [1]. Remarks (1.2): 1. Consider the Z-module M = Z8 ⊕ Z2. In this module there are eleven submodules which are <(0, 0)>, <(1, 0)>, <(0, 1)>, <(1, 1)>, <(2, 0)>, <(2, 1)>, <(4, 0)>, <(4, 1)>, <(0, 1), (4, 0)>, <(2, 0), (4, 1)>, and M. The semi-essential submodules of M are <(1, 1)>, <(1, 0)>, <(2, 0)>, <(2, 1)>, <(4, 0)>, <(0, 1), (4, 0)>, <(2, 0), (4, 1)> and M. In fact each one of them intersects with each nonzero prime submodule of M is nonzero, where the prime submodules of M are <(2, 0), (4, 1)>, <(1, 1)>, <(1, 0)>, and <(2, 0)>. 2. When a submodule N of an R-module M is nonzero in the Def (1.1), then N is a semi- essential submodule if N ∩ P ≠ (0) for each prime submodule P of M, and this is the same definition which is said by Ali and Nada in [1]. 3. Every module is a semi-essential submodule of itself. 4. For the concept of the essential submodules, (0) is an essential submodule of an R- module M if and only if M = (0), but (0) may be semi-essential submodule in a nonzero module. In fact (0) ≤sem M if and only if M has only one prime submodule which is (0), for example (0) is a semi-essential submodule of the Z-module, Z2, while (0) is not semi-essential submodule of Z. 5. The sum of two semi-essential submodules is also semi-essential submodule. Proof (5): Let M be an R-module and let L and K be two essential submodules of M. Note that L ≤ L+K, since L ≤sem M, so by [1], L+K ≤sem M. 6. Let M be an R-module, and let N ≤ M. Then for each R-module M' and for each homomorphism f: M → M' with ker f ∩ N ≠ (0), implies that N ≤sem M. 181 | Mathematics 2015) عام 1(العدد 28المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol. 28 (1) 2015 Proof (6): Let P be a nonzero prime submodule of M, and let π: M → be the natural epimorphism. By assumption ker π ∩ N ≠ (0). But ker π = P, then P ∩ N ≠ (0), hence N ≤sem M. Proposition (1.3): Let f: M → M' be an isomorphism. If N ≤sem M, then f(N) ≤sem M'. Proof: Let P be a nonzero prime submodule of M'. Since f is an epimorphism, then f-1(P) is a prime submodule of M [12, Prop. 3.8, P.10 ]. But N ≤sem M, then N ∩ f-1(P) ≠ (0), On the other hand f is a monomorphism thus f(N) ∩ P ≠ (0), and we are done. In [1], Ali and Nada gave an example verified, that the class of semi-essential submodules was didn't satisfy the transitive property for nonzero submodules. But in this work, we show that the example which they gave it in [1] is not true, and we prove that the class of semi-essential submodules satisfies the transitive property. In fact Ali and Nada said that 4 ≤sem 2 and 2 ≤sem Z12, but 4 ≰sem Z12. In fact 4 ≰sem 2 since 6 is a prime submodule of 2 and 4 ∩ 6 = (0). However, in the following proposition we give the proof of the transitive property for nonzero semi-essential submodules. Before that we need the following Lemma which appeared in [3, Prop (1.7), p.11]. Lemma (1.4): Let C be an R-module, if P is a prime submodule of C and B is a submodule of C, such that B ≰ P, then P ∩ B is a prime submodule in B. Proposition (1.5): Let A, B, C be R-modules such that A ≤ B ≤ C. Suppose that A is a nonzero submodules of M. If A ≤sem B and B ≤sem C then A ≤sem C. Proof: Let P be a prime submodule of C such that A ∩ P = (0). Note that (0) = A ∩ P = (A ∩ P) ∩ B = A ∩ (P ∩ B). But P is a prime submodule of C, so we have two cases. If B ≤ P then (0) = A ∩ (P ∩ B) = A ∩ B, hence A ∩ B = (0), but A ≤ B, so A ∩ B = A, which is implies that A = (0). But this is a contradiction with our assumption. Thus B ≰ P, and by Lemma (1.4), P ∩ B is a prime submodule of B. But A ≤sem B, therefore P ∩ B = (0), and since B ≤sem C, then P = (0), that is A ≤sem C. Remark (1.6): The condition A ≠ (0) in Prop (1.5) is necessary. In fact in the Z-module Z12, (0) is a semi-essential submodule of {0, 6} and {0, 6} is a semi-essential submodule of Z12, but (0) not semi-essential in Z12. The converse of Prop (1.5) is not true in general, as the following example shows. Example (1.7): Consider the Z-module, Z36, the submodule ( 18 ) is a semi-essential submodule of Z36. But (18) is not semi-essential submodule of (2). 2. Other results on semi-essential submodules In this section, we introduce other properties of semi-essential submodules. Recall that an R-module M is called fully prime, if every proper submodule of M is a prime submodule [5], and a nonzero R-module is called fully essential, if every nonzero semi-essential submodule of M is an essential submodule of M [11]. Tamadher in [8, Lemma 3.7], proved that if A and B are prime submodules of an R- module M and A ≤ B, then A is a prime submodule in B. In fact B need not be necessary prime submodule in M. We use this statement to prove the following proposition, which is forming a generalization of the result which was given in [11, Lemma (1.4)]. Proposition (2.1): Let M be a fully prime R-module, and let (0) ≠ N ≤ M. Then N ≤sem L if and only if N ≤e L for every submodule L of M. Proof ⇒): Assume that N is a semi essential submodule of L, and let A be a submodule of L such that N ∩ A = (0). Since M is a fully prime module then both of N and A are prime submodules of M, and by [8, Lemma 3.7] A is a prime submodule of L. But N is a semi- essential submodule of L, therefore A = (0), that is N is an essential submodule of L. ⇐): It is clear. Corollary (2.2): Every fully prime module is a fully essential module. 182 | Mathematics 2015) عام 1(العدد 28المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol. 28 (1) 2015 Recall that a nonzero R-module M is called semi-uniform if every nonzero R-submodule of M is semi-essential. A ring R is called semi-uniform if R is a semi-uniform R-module, [1]. Proposition (2.3): Let M be an R-module. Then M is uniform if and only if M is semi- uniform and fully essential. Proof ⇒): It is clear. ⇐): Let N be a nonzero submodule of M, since M is a semi-uniform module, then N ≤sem M. But M is a fully essential module, then N ≤e M. Corollary (2.4): Let M be a fully prime module, then a module M is uniform if and if M is a semi-uniform module. The following proposition appeared in [11], it deals with the direct sum of semi-essential submodules, and we give the proof for completeness. are 2and M 1module where M-be a fully prime R 2M ⨁ 1Let M = MProposition (2.5): -is a semi2 K ⊕ 1. Then K2≤ M 2K and (0) 1 ≤ M 1K submodules of M, and let (0) and 1essential submodule of M-is a semi 1if and only if K 2M ⊕ 1essential submodule of M .2essential submodule of M-is a semi 2K Proof ⇒): Since M is a fully prime module, then by [11, Lemma (1.14)] K1 ⊕ K2 is an essential submodule of M1 ⊕ M2, and by [2], K1 is an essential submodule of M1 and K2 is an essential submodule of M2. But every essential submodule is a semi-essential, so we are done. ⇐): It follows similarly. In the following proposition, we give another result for the direct sum of semi-essential submodules. Proposition (2.6): Let M = M1 ⨁ M2 be an R-module where M1 and M2 are submodules of M, and let K1 ≤ M1 and K2 ≤ M2. If K1 ⨁ K2 is a semi-essential submodule of M1 ⨁ M2, then K1 is a semi-essential submodule of M1, provided that every prime submodule of M1 is a prime submodule of M. Proof: Let P1 be a prime submodule of M1 such that K1 ∩ P1 = (0). We can easily prove that (K1 ⨁ K2) ∩ P1 = (0). By assumption P1 is a prime submodule of M and K1 ⨁ K2 ≤sem M, Thus P1 = (0). Recall that the prime radical of an R-module M is denoted by rad(M), and it is the intersection of all prime modules of M [10]. Proposition (2.7): Let M be an R-module and let (0) ≠ N ≤ M. If N' is a semi relative complement of N in M, and N' ≤ rad(M), then N ⨁ N' ≤sem M. Proof: Consider the natural epimorphism π: M → . Since N' is a semi relative complement of N in M, so by [1], ⊕ ≤sem . But ker π = N' and N' ≤ rad(M), then by [1], π-1 ( ⊕ ) ≤sem M. Hence N ⨁ N' ≤sem M. Ali and Nada in [1] showed by an example that the intersection of two semi-essential submodules need not be semi-essential submodule, and they satisfied that under certain condition, see [1]. In this work we give a deferent condition. Proposition (2.8): Let M be an R-module and let N1 and N2 be semi-essential submodules of M such that N1 ∩ N2 ≠ (0) and all prime submodules of N1 are prime submodules of M, then N1 ∩ N2 ≤sem M. Proof: Let P be a prime submodule of M such that (N1 ∩ N2) ∩ P = (0). This implies that N2 ∩ (N1∩ P) = (0). If N1 ≤ P, then we have a contradiction with the assumption, thus N1 ≰ P. This implies that N1 ∩ P is a prime submodule of N1 [3, Prop (1-7, P. 11)]. Since N2 ≤sem M 183 | Mathematics 2015) عام 1(العدد 28المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol. 28 (1) 2015 and by our assumption N1 ∩ P is a prime submodule of M, then N1 ∩ P = (0). But N1 ≤sem M, therefore P = (0), hence N1 ∩ N2 ≤sem M. Note that the condition "all prime submodules of N1 are prime submodules of M" which we used in Prop (2.8) can be applied also for N2. Proposition (2.9): Let M be an R-module, and let N1 and N2 are semi-essential submodules of M such that N2 ∩ P is a prime submodules of M for all prime submodule P of M, then N1 ∩ N2 ≤sem M. Proof: Let P be a prime submodule of M such that (N1 ∩ N2) ∩ P = (0). This implies that N1 ∩ (N2 ∩ P) = (0). But N2 ∩ P is a prime submodule of M and since N1 ≤sem M, then N2 ∩ P = (0). Moreover, since N2 ≤sem M, thus P = (0), and hence N1 ∩ N2 ≤sem M. Recall that, an R-module M is called multiplication, if for each submodule N of M, there exists an ideal I of R such that N = IM [4]. Proposition (2.10): Let M be a faithful and multiplication module such that M satisfies the condition (*), and let I, J be ideals of R. If IM ≤sem JM, then I ≤sem J. Condition (*): For any two ideals L and K of R, if L is a prime ideal of K, then LM is a prime submodule of KM. Proof: Let P be a prime ideal of J such that I ∩ P = (0), then (I ∩ P)M = (0)M. Since M is a faithful and multiplication, therefore IM ∩ PM = (0) [6, Th (1.7)]. By condition (*), PM is a prime submodule of JM. But IM ≤sem JM, then PM = (0). Since M is a faithful module so P = (0), thus I ≤sem J. The converse of Prop (2.10) is true without using the condition (*), but we need other condition as the following proposition shows. Proposition (2.11): Let M be a finitely generated, faithful and multiplication R-module. If I ≤sem J then IM ≤sem JM for every ideals I and J of R. Proof: Let P be a prime submodule of JM such that IM ∩ P = (0). Since M is a multiplication module, then P = EM for some prime ideal E of R [6, Cor (2.11)]. So IM ∩ EM = (0), this implies that (I ∩ E) M = (0). Since M is a faithful module, then I ∩ E = (0). Since EM JM and M is a finitely generated, faithful and multiplication module so by [6, Th (3.1)] E J. But E is a prime ideal of R, then E is a prime ideal of J [8, Lemma 3.7]. Since I is a semi-essential ideal of J, then E = (0), and hence P = (0). That is IM ≤sem JM. From Prop (2.10) and Prop (2.11) we have the following theorem. Theorem (2.12): Let M be a finitely generated, faithful and multiplication module such that M satisfies the condition (*). Then I ≤sem J if and only if IM ≤sem JM for every two ideals I and J of R. It is well known that If a ring R has only one maximal ideal I, then I is an essential ideal of R if and only if I ≠ (0). In the following proposition we generalize this statement in one direction to essential (hence semi-essential) submodules. Proposition (2.13): Let M be a nonzero multiplication R-module with only one maximal submodule N, if N ≠ (0). Then N is an essential (hence semi-essential) submodule of M. Proof: Let P be a submodule of M with P ⋂ N = (0). If P = M, then M ⋂ N = (0), hence N = (0) which is a contradiction. Thus P is a proper submodule of M, and since M is a nonzero multiplication module, so by [6, Th (2.5)], P contained in some maximal submodule of M. But M has only one maximal submodule which is N. Thus P ⊆ N, this implies that P = (0), that is N is an essential (hence semi-essential) submodule of M. Proposition (2.14): Let M be a finitely generated R-module with only one nonzero maximal submodule N, then N is an essential (hence semi-essential) submodule of M. Proof: In similar way, and by using [13, Prop (1.6), P. 7] instead of [6, Th (2.5)]. We end this work by the following theorem which gives the hereditary of fully essential property between R-module, M and the ring R. 184 | Mathematics 2015) عام 1(العدد 28المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol. 28 (1) 2015 Theorem (2.15): Let M be a nonzero faithful and multiplication R-module. Then M is a fully essential module if and only if R is a fully essential ring. Proof ⇒): Assume that M is a fully essential module, and let I be a nonzero semi-essential ideal of R, then IM is a submodule of M say N. This implies that N is a semi-essential submodule of M [1]. Since I (0) and M is a faithful module, then N (0). But M is a fully essential module, thus N is an essential submodule of M. Since M is a faithful and multiplication module, therefore I is an essential ideal of R [6, Th (2.13)], that is R is a fully essential ring. ⇐): Suppose that R is a fully essential ring and let (0) ≠ N ≤sem M. Since M is a multiplication module, then N = IM for some semi-essential ideal I of R. By assumption I is an essential ideal of R. But M is a faithful and multiplication module then N is an essential submodule of M [6, Th (2.13)]. That is M is a fully essential module. References [1] Ali. S. Mijbass and Nada. K. Abdullah, (2009), Semi-essential submodule and semi- uniform modules. J. of Kirkuk University-Scientific studies; 4 (1), 48-58. [2] Anderson, F.W. and Fuller, K.R, (1992), Rings and categories of modules, Springer- Verlag, New York., Academic Press Inc. London. [3] Athab, E. A., (1996), Prime and semi prime submodules M. SC. Thesis, university of Baghdad [4] Barnard, A., (1981), Multiplication modules. J. Algebra, 71: 174-178. [5] Behboodi, M., Karamzadeh, O. A. S. and Koohy, H., (2004), Modules whose certain submodule are prime, Vietnam J. of Mathematics, 32( 3): 303-317. [6] El-Bast, Z. A. and Smith, P. F., (1988), Multiplication modules, Comm. In Algebra, 16: 755-779. [7] Goodearl, K. R., (1972), Ring theory, Marcel Dekker, New York. [8] Ibrahiem, T. A., (2011), Prime extending module and S-prime module, J. of Al-Nahrain Univ., 14(4),166-170. [9] Kasch, F., (1982), Modules and rings. London: Academic Press. [10] Larsen, M. D. and McCarthy, P. J.,(1971), Multiplicative theory of ideals, Acad. press, New York and London. [11] Ahmed, M. A. and Dakheel, Sh. O., S-maximal submodules, J. of Baghdad for Science, Preprint. [12] Saymeh, S. A., (1979), On prime R-submodules, Univ. Ndc. Tucuma'n Rev. Ser. A29, 129-136. [13] Sharp, D.W. and Vamos, P., (1972), Injective modules, Cambridge Univ. Press.   185 | Mathematics 2015) عام 1(العدد 28المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol. 28 (1) 2015 حول المقاسات الجزئية شبه الجوھرية منى عباس أحمد ميساء رياض عباس جامعة بغداد –كلية العلوم للبنات –قسم الرياضيات 2015كانون الثاني 5قبل البحث في : 2014تشرين االول 20أستلم البحث في : الخالصة . ھدفنا في ھذا البحث Rمقاساً أحادياً أيسر على Mحلقة ابدالية ذات عنصر محايد, وليكن Rلتكن وھرية التي قدمھا ھو التقصي عن بعض النتائج الجديدة (على حد علمنا) حول المقاسات الجزئية شبه الج N ∩ P ≠ 0بأنه شبه جوھري، إذا كان Mمن Nالباحثان علي سبع وندى الدبان، إذ يقال للمقاس الجزئي لقد قمنا بإجراء تعديل يسير لھذا التعريف ليشمل المقاس M.من Nلكل مقاس جزئي أولي غير صفري الصفري، كما قدمنا العديد من القضايا والخواص الجديدة لھذا النوع من المقاسات الجزئية. نتظمة، المقاسات الم المقاسات الجزئية شبه الجوھرية، المقاسات الجزئية الجوھرية،الكلمات المفتاحية: المنتظمة، المقاسات األولية المتكاملة, المقاسات الجوھرية المتكاملة.المقاسات شبه