PARADIGMA BARU PENDIDIKAN MATEMATIKA DAN APLIKASI ONLINE INTERNET PEMBELAJARAN How to cite: I. Maulana, “Schanuel’s Lemma in P-Poor Modules”, mantik, vol. 5, no. 2, pp. 66-82, October 2019. Schanuel’s Lemma in P-Poor Modules Iqbal Maulana Universitas Singaperbangsa Karawang, hmiqbal1202@gmail.com doi: https://doi.org/10.15642/mantik.2019.5.2.76-82 Abstrak: Modul merupakan perumuman dari ruang vektor aljabar linier yaitu dengan memperumum lapangan skalarnya menjadi ring dengan elemen satuan. Dalam teori modul terdapat konsep modul proyektif, yaitu suatu modul atas ring R yang proyektif relatif terhadap semua modul atas R Selanjutnya, diperoleh fakta bahwa setiap modul atas R adalah modul proyektif relatif terhadap sebarang modul semisederhana atas R. Jika P adalah suatu modul atas R yang proyektif relatif hanya terhadap semua modul semisederhana atas R saja, maka P disebut modul p-miskin. Dalam pembahasan modul proyektif terdapat suatu lemma yang berkaitan dengan keekuivalenan dua buah modul K1 dan K2 dengan syarat terdapat dua buah modul proyektif P1 dan P2 sedemikian hingga 1 2K P isomorfik dengan 2 1K P . Lemma tersebut dikenal sebagai lemma Schanuel di modul proyektif. Karena modul p-poor merupakan kasus khusus dari modul proyektif, maka pada tulisan ini akan dibahas tentang lemma Schanuel di modul p-poor. Kata kunci: modul proyektif, modul semisederhana, modul p-poor, lemma Schanuel Abstract: Modules are a generalization of the vector spaces of linear algebra in which the “scalars” are allowed to be from a ring with identity, rather than a field. In module theory there is a concept about projective module, i.e. a module over ring R in which it is projective module relative to all modules over ring R. Next, there is the fact that every module over ring R is projective module relative to all semisimple modules over ring R. If P is a module over ring R which it’s projective relative only to all semisimple modules over ring R, then P is called p-poor module. In the discussion of the projective module, there is a lemma associated with the equivalence of two modules K1 and K2 provided that there are two projective modules P1 and P2 such that 1 2K P is isomorphic to 2 1K P . That lemma is known as Schanuel’s lemma in projective modules. Because the p-poor module is a special case of the projective module, then in this paper will be discussed about Schanuel’s lemma in p-poor modules. Keywords: projective module, semisimple module, p-poor module, Schanuel’s lemma Jurnal Matematika MANTIK Volume 5, Nomor 2, October 2019, pp. 76-82 ISSN: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 I Maulana Schanuel’s Lemma in P-Poor Modules 77 1. Introduction Let M and N are R-modules, i.e. modules over a ring R. In this paper, Mod-R denotes the set of all right R-modules and SSMod-R the set of all semisimple right R-modules. An R-module is called a semisimple module if that module is a direct sum of simple modules [5]. A non-zero R-module is called a simple module if that module has no non-trivial submodules. In other words, its submodule is only {0} and himself. Following [3], for any R-module M, 𝔓𝑟−1(M) = { N Mod R − | M is N - projective module} is called the projectivity domain of M. If 𝔓𝑟−1(M) = Mod- R, then M is called a projective module. Next, Alahmadi et al. [1] which discuss poor-module become the initial idea of the emergence of p-poor module concept, which p-poor module is dual of poor-module. Furthermore, this p-poor module is a special case of the projective module because the projectivity domain of p-poor only consists of all semisimple modules over ring R [2]. Regarding the existence of the p-poor module, it was found that each ring has a p-poor module. As for the formation of the p-poor module, it was found that an R-module, which is the result of the direct sum of all cyclic modules over R is a p-poor module [2]. This paper is inspired by similar ideas and problems in [4][5], where there is a lemma introduced by Stephen Schanuel in 1958 and known as the Schanuel’s lemma in projective modules. That lemma associated with the equivalence of two modules K1 and K2 provided that there are two projective modules P1 and P2 such that 𝐾1 ⊕𝑃2 is isomorphic to 𝐾2 ⊕𝑃1. The organization of this paper describes as follows: section 2 explains a basic theory about exact sequences of R-modules and semisimple module. The explanation about the Schanuel’s lemma in projective modules and Schanuel’s lemma in p-poor modules will be presented in section 3. In section 4, we conclude the discussion. 2. Basic Theory In this section, we define the external direct sum, the short exact sequence, the split exact sequence, and some properties of the semisimple module. 2.1 External Direct Sum Before we define the external direct sum, will first be discussed about the direct product. Definition 2.1. [3] The cartesian product ×𝐴 𝑋𝛼 of the sets {𝑋𝛼}𝛼∈𝐴 be the set of all A-tuple (𝑥𝛼)𝛼∈𝐴 such that 𝑥𝛼 ∈ 𝑋𝛼, for all 𝛼 ∈ 𝐴. If A is finite, 𝐴 = {1,…,𝑛} then be obtained ×𝐴 𝑋𝛼 = 𝑋1 ×…×𝑋𝑛 = {(𝑥1,…,𝑥𝑛)|𝑥𝑖 ∈ 𝑋𝑖, 𝑖 = 1,…,𝑛}. Definition 2.2. [3] Let {𝑀𝜆}𝜆∈Λ be the set of R-modules. Defined the operations in ×Λ 𝑀𝜆, for every (𝑥𝜆)𝜆∈Λ , (𝑦𝜆)𝜆∈Λ ∈ ×Λ 𝑀𝜆 and 𝑟 ∈ 𝑅 then (𝑥𝜆)𝜆∈Λ +(𝑦𝜆)𝜆∈Λ = (𝑥𝜆 +𝑦𝜆)𝜆∈Λ and 𝑟(𝑥𝜆)𝜆∈Λ = (𝑟𝑥𝜆)𝜆∈Λ . Next, the cartesian product ×Λ 𝑀𝜆, together with the above operations is R- modules. Furthermore, the module ×Λ 𝑀𝜆 is said to be the direct product of {𝑀𝜆}𝜆∈Λ and be written ∏Λ𝑀𝜆. Definition 2.3. [3] Let {𝑀𝜆}𝜆∈Λ be the set of R-modules. The external direct sum of {𝑀𝜆}𝜆∈Λ is defined as ⨁ 𝑀𝜆Λ = {𝑚 ∈ ∏Λ𝑀𝜆 | 𝜋𝜆 (𝑚) ≠ 0 𝑓𝑜𝑟 𝜆 ∈ Λ is finite}. Jurnal Matematika MANTIK Vol. 5, No. 2, October 2019, pp. 76-82 78 2.2 Exact Sequences The concept of exact sequences of R-modules and R-module homomorphisms and their relation to direct summands is a useful tool to have available in the study of modules. We start by defining exact sequences of R-modules. Definition 2.4. [6] Let R be a ring. A sequence of R-modules M and R-module homomorphisms f … 𝑓𝑖−1 → 𝑀𝑖−1 𝑓𝑖 →𝑀𝑖 𝑓𝑖+1 → 𝑀𝑖+1 𝑓𝑖+2 → … (1) is said to be exact at 𝑀𝑖 if 𝐼𝑚(𝑓𝑖) = 𝐾𝑒𝑟(𝑓𝑖+1). The sequence is said to be exact if it is exact at each 𝑀𝑖. As particular cases of Definition 2.1. note that if 𝑀, 𝑀1, and 𝑀2 are R-modules 1. 0→𝑀1 𝑓 →𝑀 is exact if and only if f is injective, 2. 𝑀 𝑔 →𝑀2 →0 is exact if and only if g is surjective, and 3. The sequence 0→𝑀1 𝑓 →𝑀 𝑔 →𝑀2 →0 (2) is exact if and only if f is injective, g is surjective and 𝐼𝑚(𝑓) = 𝐾𝑒𝑟(𝑔). Definition 2.5. [7] Given a sequence of R-modules 0→𝑀1 𝑓 →𝑀 𝑔 →𝑀2 →0 (3) 1. The sequence (3) is said to be a short exact sequence if it is exact. 2. The sequence (3) is said to be a split exact sequence (or just split) if it is exact and if 𝐼𝑚(𝑓) = 𝐾𝑒𝑟(𝑔) is a direct summand of M. Next, in the following theorem will be given a characterization of split exact sequence. Theorem 2.1. [7] If 0→𝑀1 𝑓 →𝑀 𝑔 →𝑀2 →0 (4) is a short exact sequence of R-modules, then the following are equivalent: 1. There exists a homomorphism 𝛼: 𝑀 → 𝑀1 such that 𝛼 ∘𝑓 = 𝑖𝑑𝑀1. 2. There exists a homomorphism 𝛽: 𝑀2 → 𝑀 such that 𝑔 ∘𝛽 = 𝑖𝑑𝑀2. 3. The sequence (4) is split exact. If these equivalent conditions hold then 𝑀 ≅ 𝐼𝑚(𝑓)⊕𝐾𝑒𝑟(𝛼) ≅ 𝐾𝑒𝑟(𝑔)⊕𝐼𝑚(𝛽) ≅ 𝑀1 ⊕𝑀2 2.3 Semisimple Module Next theory is needed in the next discussion is a semisimple module and some of its properties. However, it will first be defined as a simple module. I Maulana Schanuel’s Lemma in P-Poor Modules 79 Definition 2.6. [3] A non-zero R-module M is called a simple module if M has no non-trivial submodules. In other words, the submodule of M is only {0} and M. Definition 2.7. [6] An R-module M is called a semisimple module if M is a direct sum of simple modules. A semisimple module has some characterization which will be given in the following proposition. Proposition 2.2. [6] For an R-module M, the following properties are equivalent: 1. M is a semisimple module. 2. Every submodule of M is a direct summand. 3. Every exact sequence 0 → 𝐾 → 𝑀 → 𝐿 → 0 splits, for each K and L are R- modules. 3. Main Results Based on the previous introduction, we have that p-poor module is a special case of the projective module because the projectivity domain of p-poor only consists of all semisimple modules over ring R. In other words, R-modules P is p- poor if for every semisimple R-modules S satisfies for each epimorphism 𝑔 ∶ 𝑆 → 𝑁 and homomorphism 𝑓 ∶ 𝑃 → 𝑁 there exists a homomorphism ℎ ∶ 𝑃 → 𝑆 such that 𝑔 ∘ℎ = 𝑓 (i.e. the following diagram commute). Therefore, before we explain Schanuel’s lemma in p-poor modules, we will first discuss Schanuel’s lemma in projective modules. 3.1 Schanuel’s Lemma in Projective Modules This lemma associated with the equivalence of two modules M1 and M2 provided that there are two projective modules P1 and P2 such that 𝑀1 ⊕𝑃2 is isomorphic to 𝑀2 ⊕𝑃1. Furthermore, it will be discussed in the following lemma. Lemma 3.1. [4] Given the sequences of R-modules 0→𝑀1 𝑓1 →𝑃1 𝑔1 →𝑀→0 (5) 0→𝑀2 𝑓2 →𝑃2 𝑔2 →𝑀→0 (6) If (5) and (6) are exact with 𝑃1 and 𝑃2 are projective, then 𝑀1 ⊕𝑃2 is isomorphic to 𝑀2 ⊕𝑃1. Proof. From R-modules 𝑃1 and 𝑃2 can be formed a direct sum 𝑃1 ⊕𝑃2. Next, be formed 𝑋 = {(𝑝1,𝑝2) ∈ 𝑃1 ⊕𝑃2|𝑔1(𝑝1) = 𝑔2(𝑝2)}. Clearly, 𝑋 ⊆ 𝑃1 ⊕𝑃2 and 𝑋 ≠ ∅ because (0,0) ∈ 𝑋. Then, for each (𝑥1,𝑥2) and (𝑦1,𝑦2) in X and r in R, we S P Jurnal Matematika MANTIK Vol. 5, No. 2, October 2019, pp. 76-82 80 see that 𝑔1(𝑥1 +𝑦1) = 𝑔1(𝑥1)+𝑔1(𝑦1) = 𝑔2(𝑥2)+𝑔2(𝑦2) = 𝑔2(𝑥2 +𝑦2) and 𝑔1(𝑥1𝑟) = 𝑔1(𝑥1)𝑟 = 𝑔2(𝑥2)𝑟 = 𝑔2(𝑥2𝑟). So, we have (𝑥1 +𝑦1,𝑥2 +𝑦2) and (𝑥1𝑟,𝑥2𝑟) in X. In other words, X is submodule of 𝑃1 ⊕𝑃2. Next, we see that 𝑔1 is epimorphism (surjective homomorphism) so that we have 𝑀 = 𝑔1(𝑃1). Since 𝑔2 is also epimorphism, then for each 𝑔1(𝑃1) ∈ 𝑀 there exists 𝑝2 ∈ 𝑃2 such that 𝑔1(𝑝1) = 𝑔2(𝑝2). Defined homomorphism 𝜋1 : 𝑋 → 𝑃1 with 𝜋1(𝑝1,𝑝2) = 𝑝1. Then, we have 𝐾𝑒𝑟 (𝜋1) = {(𝑝1,𝑝2) | 𝜋1(𝑝1,𝑝2) = 0 } = {(𝑝1,𝑝2) | 𝑝1 = 0 } = {(0,𝑝2) | 𝑔2(𝑝2) = 0 } ≅ 𝐾𝑒𝑟 (𝑔2) = 𝐼′𝑚 (𝑓2) Furthermore, based on the particular cases of Definition 2.1, because (6) are exact, then 𝑓2 is monomorphism (injective homomorphism), and because 𝑓2 is injective, then we have 𝐼𝑚 (𝑓2) ≅ 𝑀2. As a result, we have 𝐾𝑒𝑟 (𝜋1) ≅ 𝑀2. Next, can be formed a short exact sequence 0 → 𝑀2 → 𝑋 𝜋1 → 𝑃1 → 0 (7) Since 𝑃1 is a projective module, there exists a homomorphism ℎ: 𝑃1 → 𝑋 such that 𝜋1 ∘ℎ = 𝑖𝑑𝑃1 , then the sequence (7) is split exact, and we have 𝑋 ≅ 𝑀2 ⊕𝑃1. Furthermore in an analogous way, then can be formed a short exact sequence 0 → 𝑀1 → 𝑋 𝜋2 → 𝑃2 → 0 (8) and we have 𝑋 ≅ 𝑀1 ⊕𝑃2. Therefore, we have 𝑀1 ⊕𝑃2 ≅ 𝑀2 ⊕𝑃1. 3.2 Schanuel’s Lemma in P-Poor Modules Next, can be made Schanuel’s lemma in p-poor modules, i.e. we replace sufficient conditions projective module in Lemma 3.1. with p-poor module which it is also a semisimple module, or we call that module as a semisimple p-poor. This is because the p-poor module is a special case of the projective module, where the projectivity domain of p-poor only consists of all semisimple modules. Therefore, need a certain condition is semisimple so that the concept of its projective module can be used in the p-poor module. Lemma 3.2. Given the sequences of R-modules 0→𝑀1 𝑓1 →𝑃1 𝑔1 →𝑀→0 (9) 0→𝑀2 𝑓2 →𝑃2 𝑔2 →𝑀→0 (10) If (9) and (10) are exact with 𝑃1 and 𝑃2 are semisimple p-poor modules, then 𝑀1 ⊕ 𝑃2 is isomorphic to 𝑀2 ⊕𝑃1. Proof. From semisimple p-poor modules 𝑃1 and 𝑃2 , then we have 𝑃1 ⊕𝑃2 is also semisimple p-poor module. Next, be formed 𝑊 = {(𝑝1,𝑝2) ∈ 𝑃1 ⊕ 𝑃2|𝑔1(𝑝1) = 𝑔2(𝑝2)}. Clearly, W is a submodule of 𝑃1 ⊕𝑃2 because its proof is same with the proof of X is a submodule of 𝑃1 ⊕𝑃2 in Lemma 3.1. Furthermore, according to [3] because every submodule of a semisimple module is semisimple, then we have W is a semisimple module. Next, we see that 𝑔1 is epimorphism (surjective homomorphism) so that we have 𝑀 = 𝑔1(𝑃1). Since 𝑔2 is also epimorphism, then for each 𝑔1(𝑃1) ∈ 𝑀 there exists I Maulana Schanuel’s Lemma in P-Poor Modules 81 𝑝2 ∈ 𝑃2 such that 𝑔1(𝑝1) = 𝑔2(𝑝2). Defined homomorphism 𝜋1: 𝑊 → 𝑃1 with 𝜋1(𝑝1,𝑝2) = 𝑝1. Then, we have 𝐾𝑒𝑟 (𝜋1) = {(𝑝1,𝑝2) | 𝜋1(𝑝1,𝑝2) = 0 } = {(𝑝1,𝑝2) | 𝑝1 = 0 } = {(0,𝑝2) | 𝑔2(𝑝2) = 0 } ≅ 𝐾𝑒𝑟 (𝑔2) = 𝐼𝑚 (𝑓2) Furthermore, because 𝑓2 is monomorphism (injective homomorphism), then we have 𝐼𝑚 (𝑓2) ≅ 𝑀2. As a result, we have 𝐾𝑒𝑟 (𝜋1) ≅ 𝑀2. Next, can be formed a short exact sequence 0 → 𝑀2 → 𝑊 𝜋1 → 𝑃1 → 0 (11) Since 𝑃1 is a p-poor module (i.e. projective module which its projectivity domain only consists of all semisimple modules), then for semisimple module W there exists homomorphism ℎ: 𝑃1 → 𝑊 such that 𝜋1 ∘ℎ = 𝑖𝑑𝑃1. In other words, the sequence (11) is split exact and we have 𝑊 ≅ 𝑀2 ⊕𝑃1. Furthermore in an analogous way, then can be formed a short exact sequence 0 → 𝑀1 → 𝑊 𝜋2 → 𝑃2 → 0 (12) and we have 𝑊 ≅ 𝑀1 ⊕𝑃2. Therefore, we have 𝑀1 ⊕𝑃2 ≅ 𝑀2 ⊕𝑃1. 4. Conclusion Some properties which have sufficient conditions of the projective module can be modified by replacing the projective module into the p-poor module with certain additional conditions. The result of this research only discuss how to get Schanuel's lemma in p-poor modules, i.e. with modify Schanuel’s lemma in projective modules. Its method is to replace sufficient conditions projective module on Schanuel’s lemma in projective modules with a semisimple p-poor module. This is because the p-poor module is a special case of the projective module, where the projectivity domain of p-poor only consists of all semisimple modules. Actually, this lemma also as an introduction of an equivalence relation in the p-poor module, i.e. modules 𝑀1 and 𝑀2 are equivalent if there exist semisimple p-poor modules 𝑃1 and 𝑃2 such that 𝑀1 ⊕𝑃2 is isomorphic to 𝑀2 ⊕𝑃1 . Jurnal Matematika MANTIK Vol. 5, No. 2, October 2019, pp. 76-82 82 References [1] A. N. Alahmadi, M. Alkan, and S. R. Lopez-Permouth, “Poor Modules: The Opposite of Injectivity,” Glasgow Mathematical Journal 52A, pp. 7-17, 2010. [2] C. Holston, S. R. Lopez-Permouth, and N. O. Ertas, “Rings Whose Modules Have Maximal Or Minimal Projectivity Domain,” Journal of Pure and Applied Algebra 216, pp. 673-678, 2012. [3] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules (Second Edition), New York, 1992. [4] I. Kaplansky, “Fields and Rings (Second Edition),” Chicago Lectures in Mathematics Series, pp. 165-168, 1972. [5] F D Lestari et al 2019 J. Phys.: Conf. Ser. 1211 012053 [6] R. Wisbauer, Foundations of Module and Ring Theory, Germany, 1991. [7] W. A. Adkins and S. H. Weintraub, Algebra An Approach via Module Theory, New York, 1992.