Articles
OBYEK GRUP DAN OBYEK KOGRUP DARI SEBUAH KATEGORI
MATEMATIKA Vol 13, No 2 (2010): JURNAL MATEMATIKA
Publisher : MATEMATIKA
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. A category contained a classes of objects and morphism between two object. For any chategory with initial object, terminal object, product and coproduct can defined a special object i.e group object and cogroup object. Object group obtained from cathegory object which have fulfil definition like definition a group. The cogroup object is dual from group object.
On Ï [M ]Cohereditary Modules
Jurnal ILMU DASAR Vol 12, No 2 (2011)
Publisher : Fakultas MIPA Universitas Jember
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Let R be a ring with unity and N a left Rmodule. Then N is said linearly independent to R (or N is Rlinearly independent) if there exists a monomorphism Ï : R(Î) â N . We can define a generalization of linearly independency relative to an Rmodule M. N is called Mlinearly independent if there exists a monomorphism Ï:M(Î) âN.AmoduleQiscalledMsublinearlyindependentifQisafactormoduleofamodulewhichis Mlinearly independent. The set of Msublinearly independent modules is denoted by Ï [M ]. It is easy to see that Ï [M ] is subcategory of category RMod. Furthermore, any submodule, factor module and external direct sum of module in Ï [M ] are also in Ï [M ]. A module is called Ï [M ]injective if it is Pinjective, for all modules P in Ï [M ]. Q is called Ï [M ]cohereditary if Q âÏ [M ] and any factor module of Q is Ï [M ] injective. In this paper, we study the characterization of category Ï [M ]cohereditary modules. For any Q in Ï [M ], Q is a Ï [M ]cohereditary if and only if every submodule of Qprojective module in Ï [M ] is Q projective. Moreover, Q is a Ï [M ]cohereditary if and only if every factor module of Q is a direct summand of module which contains this factor module. Also, we obtain some cohereditary properties of category Ï [M ]. There are: for any Rmodules P, Q. If Q is Pinjective and every submodule of P is Qprojective, then Q is cohereditary (1); if P is Qprojective and Q is cohereditary, then every submodule of P is Qprojective (2); a direct product of modules which Ï [M ]cohereditary is Ï [M ]cohereditary (3). The cohereditary characterization and properties of category Ï [M ] above is truly dual of characterization and properties of category Ï [M ]. Keywords : Category Ï [M ], Qprojective, Pinjective, Ï [M ]cohereditary
Pembentukan Ring Bersih Menggunakan Lokalisasi Ore
Jurnal Matematika dan Sains Vol 19 No 1 (2014)
Publisher : Institut Teknologi Bandung
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Misalkan diberikan sebarang ring R (tidak harus komutatif) dan himpunan multiplikatif S Í R yang tidak memuat elemen nol. Lokalisasi Ore merupakan salah satu teknik pembentukan ring sehingga setiap elemen S memiliki invers di ring yang baru. Ring hasil lokalisasi tidak selalu mempertahankan sifat ring awal. Suatu ring sebarang dapat disisipkan ke ring bersih, ring bersihn dan ring peralihan. Pada paper ini akan dikaji sifatsifat yang diperlukan untuk menyisipkan sebarang ring ke ring tersebut menggunakan lokalisasi. Kata kunci : Lokalisasi Ore, Elemen Satuan, Ring Bersih, Ring Peralihan, Ring Bersihn. Construction of Clean Ring using Ore Localization Abstract Let R be any ring (can be non commutative) and S Í R is a multiplicative set that does not contain any zero element. Ore localization is a powerful technique to construct a universal Sinverting ring. However the localization results do not always inherit properties of the first ring. An arbitrary ring can be inserted into the clean ring, nclean ring, and exchange ring. Here, we show properties needed to insert any ring to the ring using localization. Keywords: Ore Localization, Unity, Clean Ring, Exchange Ring, nClean Ring.
MODUL Ï[M]INJEKTIVE
Journal of Mathematics and Mathematics Education Vol 1, No 2 (2011): Journal of Mathematics and Mathematics Education
Publisher : Journal of Mathematics and Mathematics Education
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AbstractÂ LetÂ R be a ring with unit and letÂ N be a left Rmodule. Then N is said linearly independent toÂ R (or N is Rlinearly independent) if there is monomorphisma Â By the definition of Rlinearly independent, we may be able to generalize linearly independent relative to the Rmodule M. Module N is said Mlinearly independent if there is monomorphisma .The module Q is said Msublinearly independent if Q is a factor module of modules which isÂ Mlinearly independent. The set of modules Msublinearly independent denoted by Â Can be shown easily that Â is a subcategory of the category RMod. Also it can be shown that the submodules, factor modules and external direct sum of modules in Â is also in the .The module Q is called Pinjective if for any morphisma Q defined on L submodules of P can be extended to morphisma Q with , where Â is the natural inclusion mapping. The module Q is called injective if Q is Pinjective, for all modules P in .In this paper, we studiet the properties and characterization of injective. Trait among others that the direct summand of a module that is injective also injective. A module is injective if and only if the direct product of these modules also are injective.Â Key words : Q ()projective, P ()injective.
Konstruksi Ring Bersih dari Sebarang Ring
Jurnal Matematika Vol 5, No 2 (2015)
Publisher : Jurnal Matematika
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The aims of this research was to construct a clean ring from any ring.Â The base on the factÂ that the endomorphism ring of every pureinjective module is clean, it was constructed a clean ring from any ring. So, the result of this research was it always could be constructed a clean ring from any ring.
Setiap Modul merupakan Submodul dari Suatu Modul Bersih
Jurnal Matematika Integratif Volume 11 No 1 (April 2015)
Publisher : Jurnal Matematika Integratif
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Diberikan ring R dengan elemen satuan. Suatu ring R dikatakan bersih apabila setiap elemennya dapat dinyatakan dalam bentuk jumlahan suatu elemen unit dan suatu elemen idempoten dari ring R, sedangkan suatu Rmodul M dikatakan bersih apabila ring endomorfismanya merupakan ring bersih. Berdasarkan sifat bahwa modul kontinu merupakan modul bersih, dalam penelitian ini ditunjukkan bahwa setiap modul merupakan submodul dari suatu modul bersih.
On Ï [M ]Cohereditary Modules
Jurnal ILMU DASAR Vol 12 No 2 (2011)
Publisher : Fakultas Matematika dan Ilmu Pengetahuan Alam Universitas Jember
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Let R be a ring with unity and N a left Rmodule. Then N is said linearly independent to R (or N is Rlinearly independent) if there exists a monomorphism Ï : R(Î) â N . We can define a generalization of linearly independency relative to an Rmodule M. N is called Mlinearly independent if there exists a monomorphism Ï:M(Î) âN. Amodule Â iscalled Msublinearly independentif Â is a factormodule of a module which isÂ Mlinearly independent. The set of Msublinearly independent modules is denoted by Ï [M ]. It is easy to see that Ï [M ] is subcategory of category RMod. Furthermore, any submodule, factor module and external direct sum of module in Ï [M ] are also in Ï [M ]. A module is called Ï [M ]injective if it is Pinjective, for all modules P in Ï [M ]. Q is called Ï [M ]cohereditary if Q âÏ [M ] and any factor module of Q is Ï [M ]injective. In this paper, we study the characterization of category Ï [M ]cohereditary modules. For any Q in Ï [M ], Q is a Ï [M ]cohereditary if and only if every submodule of Qprojective module in Ï [M ] is Qprojective. Moreover, Q is a Ï [M ]cohereditary if and only if every factor module of Q is a direct summand of module which contains this factor module. Also, we obtain some cohereditary properties of category Ï [M ]. There are: for any Rmodules P, Q. If Q is Pinjective and every submodule of P is Qprojective, then Q is cohereditary (1); if P is Qprojective and Q is cohereditary, then every submodule of P is Qprojective (2); a direct product of modules which Ï [M ]cohereditary is Ï [M ]cohereditary (3). The cohereditary characterization and properties of category Ï [M ] above is truly dual of characterization and properties of category Ï [M ].
ON FREE IDEALS IN FREE ALGEBRAS OVER A COMMUTATIVE RING
Journal of the Indonesian Mathematical Society Volume 21 Number 1 (April 2015)
Publisher : IndoMS
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Let A be a free Ralgebra where R is a unital commutative ring. An ideal I in A is called a free ideal if it is a free Rsubmodule with the basis contained in the basis of A. The denition of free ideal and basic ideal in the free Ralgebra are equivalent. The free ideal notion plays an important role in the proof of some special properties of a basic ideal that can characterize the free Ralgebra. For example, a free Ralgebra A is basically semisimple if and only if it is a direct sum of minimal basic ideals in A: In this work, we study the properties of basically semisimple free Ralgebras.DOI :Â http://dx.doi.org/10.22342/jims.21.1.170.5969
ON JOINTLY PRIME RADICALS OF (R,S)MODULES
Journal of the Indonesian Mathematical Society Volume 21 Number 1 (April 2015)
Publisher : IndoMS
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Let $M$ be an $(R,S)$module. In this paper a generalization of the msystem set of modules to $(R,S)$modules is given. Then for an $(R,S)$submodule $N$ of $M$, we define $sqrt[(R,S)]{N}$ as the set of $ain M$ such that every msystem containing $a$ meets $N$. It is shown that $sqrt[(R,S)]{N}$ is the intersection of all jointly prime $(R,S)$submodules of $M$ containing $N$. We define jointly prime radicals of an $(R,S)$module $M$ as $rad_{(R,S)}(M)=sqrt[(R,S)]{0}$. Then we present some properties of jointly prime radicals of an $(R,S)$module.DOI :Â http://dx.doi.org/10.22342/jims.21.1.199.2534
ON FREE PRODUCT OF NCOGROUPS
Journal of the Indonesian Mathematical Society Volume 18 Number 2 (October 2012)
Publisher : IndoMS
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looked at pdf abstractDOI :Â http://dx.doi.org/10.22342/jims.18.2.116.101111