Indah Emilia Wijayanti
Jurusan Matematika FMIPA Universitas Gadjah Mada, Yogyakarta

Published : 13 Documents
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OBYEK GRUP DAN OBYEK KOGRUP DARI SEBUAH KATEGORI

MATEMATIKA Vol 13, No 2 (2010): JURNAL MATEMATIKA
Publisher : MATEMATIKA

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Abstract

. 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 τ [M ]-Cohereditary Modules

Jurnal ILMU DASAR Vol 12, No 2 (2011)
Publisher : Fakultas MIPA Universitas Jember

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Abstract

Let R be a ring with unity and N a left R-module. Then N is said linearly independent to R (or N is R-linearly independent) if there exists a monomorphism φ : R(Λ) → N . We can define a generalization of linearly independency relative to an R-module M. N is called M-linearly independent if there exists a monomorphism φ:M(Λ) →N.AmoduleQiscalledM-sublinearlyindependentifQisafactormoduleofamodulewhichis M-linearly independent. The set of M-sublinearly independent modules is denoted by τ [M ]. It is easy to see that τ [M ] is subcategory of category R-Mod. Furthermore, any submodule, factor module and external direct sum of module in τ [M ] are also in τ [M ]. A module is called τ [M ]-injective if it is P-injective, for all modules P in τ [M ]. Q is called τ [M ]-cohereditary if Q ∈τ [M ] and any factor module of Q is τ [M ]- injective. In this paper, we study the characterization of category τ [M ]-cohereditary modules. For any Q in τ [M ], Q is a τ [M ]-cohereditary if and only if every submodule of Q-projective module in τ [M ] is Q- projective. Moreover, Q is a τ [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 τ [M ]. There are: for any R-modules P, Q. If Q is P-injective and every submodule of P is Q-projective, then Q is cohereditary (1); if P is Q-projective and Q is cohereditary, then every submodule of P is Q-projective (2); a direct product of modules which τ [M ]-cohereditary is τ [M ]-cohereditary (3). The cohereditary characterization and properties of category τ [M ] above is truly dual of characterization and properties of category τ [M ]. Keywords : Category τ [M ], Q-projective, P-injective, τ [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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Abstract

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 bersih-n dan ring peralihan. Pada paper ini akan dikaji sifat-sifat yang diperlukan untuk menyisipkan sebarang ring ke ring tersebut menggunakan lokalisasi. Kata kunci : Lokalisasi Ore, Elemen Satuan, Ring Bersih, Ring Peralihan, Ring Bersih-n.   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 S-inverting ring. However the localization results do not always inherit properties of the first ring. An arbitrary ring can be inserted into the clean ring, n-clean 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, n-Clean Ring.

MODUL τ[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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Abstract

Abstract Let  R be a ring with unit and let  N be a left R-module. Then N is said linearly independent to  R (or N is R-linearly independent) if there is monomorphisma  By the definition of R-linearly independent, we may be able to generalize linearly independent relative to the R-module M. Module N is said M-linearly independent if there is monomorphisma .The module Q is said M-sublinearly independent if Q is a factor module of modules which is  M-linearly independent. The set of modules M-sublinearly independent denoted by  Can be shown easily that  is a subcategory of the category R-Mod. Also it can be shown that the submodules, factor modules and external direct sum of modules in  is also in the .The module Q is called P-injective if for any morphisma Q defined on L submodules of P can be extended to morphisma Q with , where  is the natural inclusion mapping. The module Q is called -injective if Q is P-injective, 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. 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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Abstract

The aims of this research was to construct a clean ring from any ring.  The base on the fact  that the endomorphism ring of every pure-injective 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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Abstract

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 R-modul 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 τ [M ]-Cohereditary Modules

Jurnal ILMU DASAR Vol 12 No 2 (2011)
Publisher : Fakultas Matematika dan Ilmu Pengetahuan Alam Universitas Jember

Show Abstract | Original Source | Check in Google Scholar | Full PDF (251.967 KB)

Abstract

Let R be a ring with unity and N a left R-module. Then N is said linearly independent to R (or N is R-linearly independent) if there exists a monomorphism φ : R(Λ) → N . We can define a generalization of linearly independency relative to an R-module M. N is called M-linearly independent if there exists a monomorphism φ:M(Λ) →N. Amodule  iscalled M-sublinearly independentif  is a factormodule of a module which is M-linearly independent. The set of M-sublinearly independent modules is denoted by τ [M ]. It is easy to see that τ [M ] is subcategory of category R-Mod. Furthermore, any submodule, factor module and external direct sum of module in τ [M ] are also in τ [M ]. A module is called τ [M ]-injective if it is P-injective, for all modules P in τ [M ]. Q is called τ [M ]-cohereditary if Q ∈τ [M ] and any factor module of Q is τ [M ]-injective. In this paper, we study the characterization of category τ [M ]-cohereditary modules. For any Q in τ [M ], Q is a τ [M ]-cohereditary if and only if every submodule of Q-projective module in τ [M ] is Q-projective. Moreover, Q is a τ [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 τ [M ]. There are: for any R-modules P, Q. If Q is P-injective and every submodule of P is Q-projective, then Q is cohereditary (1); if P is Q-projective and Q is cohereditary, then every submodule of P is Q-projective (2); a direct product of modules which τ [M ]-cohereditary is τ [M ]-cohereditary (3). The cohereditary characterization and properties of category τ [M ] above is truly dual of characterization and properties of category τ [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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Abstract

Let A be a free R-algebra where R is a unital commutative ring. An ideal I in A is called a free ideal if it is a free R-submodule with the basis contained in the basis of A. The denition of free ideal and basic ideal in the free R-algebra 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 R-algebra. For example, a free R-algebra 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 R-algebras.DOI : http://dx.doi.org/10.22342/jims.21.1.170.59-69

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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Abstract

Let $M$ be an $(R,S)$-module. In this paper a generalization of the m-system 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 m-system 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 : http://dx.doi.org/10.22342/jims.21.1.199.25-34

ON FREE PRODUCT OF N-COGROUPS

Journal of the Indonesian Mathematical Society Volume 18 Number 2 (October 2012)
Publisher : IndoMS

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Abstract

looked at pdf abstractDOI : http://dx.doi.org/10.22342/jims.18.2.116.101-111