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Diberikan grup berhingga G dan lapangan F . Aljabar grup F [G] merupakan suatu ring sekaligus merupakan ruang vektor atas F . Diketahui, jika ruang vektor V atas F merupakan modul atas aljabar grup F [G] maka selalu dapat dikonstruksi suatu representasi ring F [G] terhadap V , yakni suatu homomorfisma ring dari F [G] ke ring semua tranformasi linear pada V . Lebih lanjut, diketahui juga F [G] dan ring semua transformasi linear pada V merupakan koaljabar atas F . Berdasarkan hal ini, jika suatu ruang vektor atas F merupakan komodul atas F [G] maka muncul permasalahan apakah dapat dikonstruksi suatu homomorfisma koaljabar dari F [G] ke koajabar semua transformasi linear pada ruang vektor tersebut. Oleh karena itu, pada tulisan ini akan diberikan pengkonstruksian homomorfisma koajabar F [G] terhadap suatu ruang vektor atas F . Selanjutnya, homomorfisma koaljabar F [G] disebut korepresentasi koaljabar F [G] terhadap suatu ruang vektor atas F .

Diberikan ring komutatif R dengan elemen satuan. Ring R disebut bersih jika setiap elemennya dapat dinyatakan sebagai jumlahan elemen unit dan idempoten. Sebuah R-modul M disebut modul bersih jika ring endomorfisma dari M adalah ring bersih. Misalkan C adalah R-koaljabar yang bersifat koasosiatif dan kounital. Sebuah C-komodul M dikatakan bersih jika ring endomorfisma dari C-komodul M adalah ring bersih. Oleh karena setiap koaljabar merupakan komodul atas dirinya sendiri, koaljabar bersih didefinisikan sebagai kejadian khusus dari komodul bersih. Pada paper ini akan dilihat keterkaitan sifat bersih pada struktur ring, modul, koaljabar dan komodul.

Let R be a ring, K,M be R-modules, L a uniserial R-module, and X a submodule of L. The triple (K,L,M) is said to be X-sub-exact at L if the sequence K→X→M is exact. Let σ(K,L,M) is a set of all submodules Y of L such that (K,L,M) is Y -sub-exact. The sub-exact sequence is a generalization of an exact sequence. We collect all triple (K,L,M) such that (K,L,M) is an X-sub exact sequence, where X is a maximal element of σ(K,L,M). In a uniserial module, all submodules can be compared under inclusion. So, we can find the maximal element of σ(K,L,M). In this paper, we prove that the set σ(K,L,M) form a category, and we denoted it by CL. Furthermore, we prove that CY is a full subcategory of CL, for every submodule Y of L. Next, we show that if L is a uniserial module, then CL is a pre-additive category. Every morphism in CL has kernel under some conditions. Since a module factor of L is not a submodule of L, every morphism in a category CL does not have a cokernel. So, CL is not an abelian category. Moreover, we investigate a monic X-sub-exact and an epic X-sub-exact sequence. We prove that the triple (K,L,M) is a monic X-sub-exact if and only if the triple Z-modules ( , , ) is a monic -sub-exact sequence, for all R-modules N. Furthermore, the triple (K,L,M) is an epic X-sub-exact if and only if the triple Z-modules ( , , ) is a monic -sub-exact, for all R-module N.

Let [Formula: see text] be a ring of the Morita context. We define a concept of Dubrovin valuation modules for bi-modules [Formula: see text] and [Formula: see text] and give necessary and sufficient conditions for [Formula: see text] to be a Dubrovin valuation ring in terms of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].

UDC 512.5Smith in paper [Mapping between module lattices, Int. Electron. J. Algebra, 15, 173–195 (2014)] introduced maps between the lattice of ideals of a commutative ring and the lattice of submodules of an -module i.e., and mappings.The definitions of the maps were motivated by the definition of multiplication modules.Moreover, some sufficient conditions for the maps to be a lattice homomorphisms are studied.In this work we define a class of -modules and observe the properties of the class. We give a sufficient conditions for the module and the ring such that the class is a hereditary pretorsion class.

Jordan ring is one example of the non-associative rings. We can construct a Jordan ring from an associative ring by defining the Jordan product. In this paper, we discuss the properties of non-associative rings by studying the properties of the Jordan rings. All of the ideals of a non-associative ring R are non-associative, except the ideal generated by the associator in R. Hence, a quotient ring can be constructed, where is the ideal generated by associators in R. The fundamental theorem of the homomorphism ring can be applied to the non-associative rings. By a little modification, we can find that is isomorphic to . Furthermore, we define a module over a non-associative ring and investigate its properties. We also give some examples of such modules. We show if M is a module over a non-associative ring R, then M is also a module over if is contained in the annihilator of R. Moreover, we define the tensor product of modules over a non-associative ring. The tensor product of the modules over a non-associative ring is commutative and associative up to isomorphism but not element by element.

Penyediaan bibit lamtoro (Leucaena leucocephala) yang berkualitas dan cepat dapat dilakukan melaluipembiakkan in vitro atau kultur jaringan. Aklimatisasi merupakan tahapan akhir dari perbanyakan tanamanpada teknik kultur jaringan. Penelitian ini bertujuan untuk membandingkan pertumbuhan tanaman lamtoroberakar dan tidak berakar serta mengetahui pengaruh berbagai komposisi media tanam terhadap pertumbuhantanaman lamtoro (Leucaena leucocephala). Rancangan yang digunakan pada penelitian ini yaitu rancanganacak lengkap faktorial 2 × 4. Faktor pertama adalah T1 (berakar) dan T2 (tidak berakar). Faktor keduaadalah M1(zeolit), M2 (zeolit, arang sekam, pupuk kandang), M3 (pasir, arang sekam, pupuk kandang), M4(tanah, pasir, arang sekam, pupuk kandang). Berdasarkan hasil percobaan, sifat tumbuh tanaman berakardengan media tanam zeolit (T1M1) memberikan respon terbaik terhadap viabilitas, adaptasi daun, warnadaun, pertambahan tinggi tanaman, serta jumlah daun dan ranting.Kata kunci: aklimatisasi, kultur jaringan, lamtoro, media tanam, sifat tumbuh perakaran

Assume that G(V,E) is a graph with V and E as its vertex and edge sets, respectively. We have G is simple, connected, and undirected. Given a function λ from a union of V and E into a set of k-integers from 1 until k. We call the function λ as a totally irregular total k-labeling if the set of weights of vertices and edges consists of different numbers. For any u ∈ V, we have a weight wt(u)=λ(u)+ ∑{uy ∈ E} λ(uy). Also, it is defined a weight wt(e)= λ(u)+ λ(uv) + λ(v) for each e=uv ∈ E. A minimum k used in k-total labeling λ is named as a total irregularity strength of G, symbolized by ts(G). We discuss results on ts of some caterpillar graphs in this paper. The results are ts(S{p,2,2,q}) = ⌈ (p+q-1)/2 ⌉ for p, q greater than or equal to 3, while ts(S{p,2,2,2,p}) = ⌈(2p-1)/2 ⌉, p ≥ 4.

Let M be a finitely generated torsion-free module over a Dedekind domain R and M[X] be a polynomial module over R[X], where X is an indeterminate. It is shown that M and M[X] are both generalized D...

In the development of Theory Radical of Rings, there are two kinds of radical constructions. The first radical construction is the lower radical construction and the second one is the upper radical construction. In fact, the class π of all prime rings forms a special class and the upper radical class of forms a radical class which is called the prime radical. An upper radical class which is generated by a special class of rings is called a special radical class. On the other hand, we also have the class of all semiprime rings which is weakly special class of rings. Moreover, we can construct a special class of modules by using a given special class of rings. This condition motivates the existence of the question how to construct weakly special class modules by using a given weakly special class of rings. This research is a qualitative research. The results of this research are derived from fundamental axioms and properties of radical class of rings especially on special and weakly special radical classes. In this paper, we introduce the notion of a weakly special class of modules, a generalization of the notion on a special class of modules based on the definition of semiprime modules. Furthermore, some properties and examples of weakly special classes of modules are given. The main results of this work are the definition of a weakly special class of modules and their properties.

Let $V$ be a submodule of a direct sum of some elements in $\mathcal{U}$, and $X$ be a submodule of a direct sum of some elements in $\mathcal{N}$, where $\mathcal{U}$ and $\mathcal{N}$ are families of $R$-modules. A $\mathcal{U}$-free module is a generalization of a free module. According to the definition of $\mathcal{U}$-free module, we define three kinds of projective$_{\mathcal{U}}$ in this research, i.e., projective$_{\underline{\mathcal{U}}}$, projective$_{\mathcal{U}}$ module, and strictly projective$_{\mathcal{U}}$ module. The notion of strictly projective$_{\mathcal{U}}$ is a generalization of the projective module. In this research, we discuss the relationship between projective modules and the three types of modules. Furthermore, we show that the properties of $\mathcal{U}$ impact the properties of the projective$_{\mathcal{U}}$ module so that we can determine some properties of the projective$_{\mathcal{U}}$ module based on the properties of the family of $\mathcal{U}$ of $R$-modules.

Let [Formula: see text] be a torsion-free module over an integral domain [Formula: see text] with quotient field [Formula: see text]. We define a concept of completely integrally closed modules in order to study Krull modules. It is shown that a Krull module [Formula: see text] is a [Formula: see text]-multiplication module if and only if [Formula: see text] is a maximal [Formula: see text]-submodule and [Formula: see text] for every minimal prime ideal [Formula: see text] of [Formula: see text]. If [Formula: see text] is a finitely generated Krull module, then [Formula: see text] is a Krull module and [Formula: see text]-multiplication module. It is also shown that the following three conditions are equivalent: [Formula: see text] is completely integrally closed, [Formula: see text] is completely integrally closed, and [Formula: see text] is completely integrally closed.

Let $R = \oplus_{n \in \Z} R_{n}$ be a strongly graded ring of type $\Z$ and $R_{0}$ is a prime Goldie ring. It is shown that the following three conditions are equivalent: (i) $R_{0}$ is a $\Z$-invariant Krull ring, (ii) $R$ is a Krull ring and (iii) $R$ is a graded Krull ring. We completely describe all $v$-invertible $R$-ideals in $Q$, where $Q$ is a quotient ring of $R$.&nbsp

Diberikan ring S dan ideal kanan A. Dapat dibangun subring terbesar di S sedemikian sehingga A merupakan ideal dua sisi di subring tersebut. Struktur ini disebut pembuat ideal dan dinotasikan dengan R=I_S(A). Di dalam paper ini akan dibahas pembuat ideal dari ideal kanan isomaksimal generatif yang selanjutnya disebut pembuat ideal mendasar. Modul atas ring S dapat dipandang sebagai modul atas subring R. Modul atas ring S dapat dipandang sebagai modul atas subring R. Untuk sebarang subring R, beberapa sifat modulnya seperti sifat proyektif dan sifat sederhana tidak diawetkan ketika mengganti ring tumpuannya. Lebih lanjut, ditunjukkan bahwa jika R pembuat ideal mendasar, sifat-sifat ini diawetkan.Let S be a ring and A be a right ideal. We can construct the largest subring of S in which A be two sided ideal. This structure is called idealizer and denoted by R=I_S(A). In this paper we discuss idealizer of generative isomaximal right ideals that is called basic idealizer. Module over S can be viewed as modules over subring R. For arbitrary subrings R, some properties of modules such as projective dan simple are not preserved between when changing the ring. Furthermore, we show that under condition that R is basic idealizer, these properties are preserved.

Let \(R\) be a commutative ring with multiplicative identity and \(P\) is a finitely generated projective \(R\)-module. If \(P^{\ast}\) is the set of \(R\)-module homomorphism from \(P\) to \(R\), then the tensor product \(P^{\ast}\otimes_{R}P\) can be considered as an \(R\)-coalgebra. Furthermore, \(P\) and \(P^{\ast}\) is a comodule over coalgebra \(P^{\ast}\otimes_{R}P\). Using the Morita context, this paper give sufficient conditions of clean coalgebra \(P^{\ast}\otimes_{R}P\) and clean \(P^{\ast}\otimes_{R}P\)-comodule \(P\) and \(P^{\ast}\). These sufficient conditions are determined by the conditions of module \(P\) and ring \(R\).

Let [Formula: see text] be a finitely generated torsion-free module over a generalized Dedekind domain [Formula: see text]. It is shown that if [Formula: see text] is a projective [Formula: see text]-module, then it is a generalized Dedekind module and [Formula: see text]-multiplication module. In case [Formula: see text] is Noetherian it is shown that [Formula: see text] is either a generalized Dedekind module or a Krull module. Furthermore, the polynomial module [Formula: see text] is a generalized Dedekind [Formula: see text]-module (a Krull [Formula: see text]-module) if [Formula: see text] is a generalized Dedekind module (a Krull module), respectively.

Diberikan ring R dan S, suatu (R,S)−bimodul V , dan suatu (S,R)−bimodul W. Misalkan T merupakan ring konteks Morita dengan komponen penyusunnya adalah R,V,W, dan S. Q(R) ring hasil bagi R, Q(S) ring hasil bagi S, Q(V ) modul hasil bagi V , dan Q(W) modul hasil bagi W. Dalam penelitian ini, ring T diasumsikan sebagai ring Goldie prima dengan ring hasil bagi dari T adalah Q(T) dengan komponen penyusunnya adalah Q(R),Q(V),Q(W), dan Q(S). Dengan kata lain, T merupakan order dalam suatu ring Artin sederhana Q(T). Lebih lanjut, akan diberikan beberapa syarat perlu dan cukup ring T merupakan order maksimal dan order Asano.

Untuk sebarang ruang topologis $X$ dapat dibentuk $Cov_X$ yaitu kategori \linebreak ruang penutup $X$ yang terhubung lintasan. Pada tulisan ini akan dibahas syarat perlu dan cukup eksistensi morfisma antara dua ruang penutup yang terhubung lintasan lokal. Untuk sebarang $x_0 \in X$ dan grup fundamental $G=\pi_1(X,x_0)$, dapat dibentuk kategori $SetG$, yaitu kategori semua himpunan yang dilengkapi aksi kanan oleh $G$. Selanjutnya dibentuk fungtor $F$ dari $Cov_X$ ke $SetG$. Dalam tulisan dibuktikan bahwa $F$ bersifat \textit{fully faithful} jika $X$ terhubung lintasan dan terhubung lintasan lokal. Akibatnya untuk mengidentifikasi morfisma-morfisma antara dua obyek $A$ dan $B$ di $Cov_X$ dapat dilakukan dengan cara melihat sifat morfisma-morfisma antara $F(A)$ dan $F(B)$. (For any topological space $X$, we can construct the category of path \linebreak connected covering spaces of $X$, denoted by $Cov_X$. In this paper we study a sufficient and necesarry condition for the existence of morphism between two locally path \linebreak connected covering spaces. For every $x_0 \in X$ and fundamental group $G=\pi_1(X,x_0)$, we can construct the category of sets with right action of $G$, denoted by $SetG$. \linebreak Furthermore, we can define a functor $F$ from $Cov_X$ to $SetG$. We proof that the functor $F$ is fully faithul if $X$ is path connected and locally path connected. From this result, we can identify morphisms between $A$ and $B$ in $Cov_X$ by using the properties of morphisms between $F(A)$ and $F(B)$. )

Let S be a ring and A be a right ideal. We can construct the largest subring of S in which A be two sided ideal. This structure is called idealizer and denoted by R = ⅡS(A). Spesifically, when A is a generative isomaximal right ideal, the idealizer is called a basic idealizer. We investigate the relationship between basic idealizer R and S and conclude that S is a minimal extension of R. Furthermore, we show some examples that for arbitrary subring R, some properties such as Noetherian, Artinian and hereditary are not necessarily preserved between R and S. We prove that under condition that R is a basic idealizer, these properties are preserved. We prove it by using module theory approach.

Let C be an R-coalgebra. We defined a tidy clean C-comodule M if M is a clean module over the dual algebra of C. Since every coalgebra C is a comodule over itself, a clean R-coalgebra C is a clean comodule over C. The cleanness of coalgebra C depends on its dual. In this paper, we have some properties of clean coalgebras based on the relationship between the cleanness in R-coalgebra C with the cleanness of its opposite, a clean coproduct of the family of clean coalgebras and a clean coalgebra from a dual algebra.

Let U be a nonempty set of R-modules and V be a submodule of ⊕_ʌU_λ, U_λ∈U for allλ∈Λ. A U_V generated module is a generalization of U-generated module by using the concept of V-coexact sequence. We say that an R-module N is generated by U_V if there is an epimorphism from V to N. In this paper, we introduce the definition of rank of U_V-generated modules. Furthermore, we investigate some properties of rank of UV-generated modules.

In this paper we give a further study on fully prime submodules. For any fully prime submodules we define a product called $\am$-product. The further investigation of fully prime submodules in this work, i.e. the fully m-system and fully prime radicals, is related to this product. We show that the fully prime radical of any submodules can be characterize by the fully m-system. As a special case, the fully prime radical of a module $M$ is the intersection of all minimal fully prime submodules of $M$.

A ring A is called a *-ring if A is a prime ring and A has no nonzero proper prime homomorphic image. The *-ring was introduced by Korolczuk in 1981. Since *-rings have an important role in radical theory of rings, the properties of *-ring have been being investigated intensively. Since every ring can be viewed as a module over itself, the generalization of *-ring into module theory is an interesting investigation. We would like to present the generalization of *-rings in module theory named *p-modules. An A-module M is called a -module if M is a prime A-module and M has no nonzero proper prime submodule. According to the result of our investigation, we show that every *-ring is a *p-module over itself. Furthermore, let A be a ring, let M be an A-module, and let I be an ideal of A with I subset (0:M)A where (0:M)A={a in A| aM={0}}. We show that M is a *p-module over A if and only if M is a *p-module over A/I. On the other hand, the essential closure *k of the class of all *-rings is a special class of rings. As the last result of our investigation, we present the special class of modules determined by *k.

Diberikan modul $M$ dan $N$, $M$ dikatakan $N$-subinjketif jika untuk setiap perluasan $K$ dari $N$, setiap homomorfisma $f:N\to M$ dapat diperluas menjadi homomorfisma $g:K\to M$. Daerah subinjektifitas suatu modul $M$ merupakan koleksi seluruh modul $N$ sehingga $M$ merupakan $N$-subinjektif. Modul \textit{indigent} merupakan modul yang daerah subinjektifitasnya hanya terdiri dari modul-modul injektif saja. Pada tulisan ini dibahas mengenai beberapa sifat tentang daerah subinjektifitas suatu modul dan beberapa sifat dari modul \textit{indigent}. Diberikan pula beberapa contoh ring yang memiliki modul \textit{indigent}}.

The aim of this paper is to bring together the properties of quasi-exact sequences and homomorphism functors. It is of interest to know whether the left quasi-exact sequences have some interesting properties related to the functors and vice versa. Moreover, we also define the dualization of left quasi-exact sequences, i.e. right quasi-coexact sequences. Our proofs show that the quasi-exactness of sequences is preserved by any homomorphism functors.

Additive code is a generalization of linear code. It is defined as subgroup of a finite Abelian group. The definitions of Hamming distance, Hamming weight, weight distribution, and homogeneous weight distribution in additive code are similar with the definitions in linear code. Different with linear code where the dual code is defined using inner product, additive code using theories in group to define its dual code because in group theory we do not have term of inner product. So, by this thesis, the definitions of dual code in additive code will be discussed. Then, this thesis discuss about a familiar theorem in dual code theory, that is MacWilliams Identity. Next, this thesis discuss about how to proof of MacWilliams Identity on adiitive code using dual codes which are defined.

Let R be a commutative ring with identity and M be a comodule over R-coalgebra C. It was already well-known that any C-comodule M is a module over dual algebra C* where C* is the set of all R-module homomorphisms from C to R. Furthermore, the category of comodule is a subcategory of the category of C*-module. Hence, any C-subcomodule of M is a C*-submodule of M, and the conversely is not true. For any non zero element m in M, C*m is a C*-submodule of M. In general, C*m is not to become a C-subcomodule of M. By using the theory of exact sequences in modules and the theory of categories, we give a condition such that C*m to be a C-subcomodule of M.

A special atom (respectively, supernilpotent atom) is a minimal element of the lattice $\mathbb{S}$ of all special radicals (respectively, a minimal element of the lattice $\mathbb{K}$ of all supernilpotent radicals). A semiprime ring $R$ is called prime essential if every nonzero prime ideal of $R$ has a nonzero intersection with each nonzero two-sided ideal of $R$. We construct a prime essential ring $R$ such that the smallest supernilpotent radical containing $R$ is not a supernilpotent atom but where the smallest special radical containing $R$ is a special atom. This answers a question put by Puczylowski and Roszkowska.

In ring and module theory, the cleanness property is well established. If any element of R can be expressed as the sum of an idempotent and a unit, then R is said to be a clean ring. Moreover, an R-module M is clean if the endomorphism ring of M is clean. We study the cleanness concept of coalgebra and comodules as a dualization of the cleanness in rings and modules. Let C be an R-coalgebra and M be a C-comodule. Since the endomorphism of C-comodule M is a ring, M is called a clean C-comodule if the ring of C-comodule endomorphisms of M is clean. In Brzezi´nski and Wisbauer (2003), the group ring R[G] is an R-coalgebra. Consider M as an R[G]-comodule. In this paper, we have investigated some sucient conditions to make M a clean R[G]-comodule, and have shown that every G-graded module M is a clean R[G]-comodule if M is a clean R-module.

An 𝑅-module 𝑀 is said to be indigent if its subinjectivity domain consists of only an injective module. In this paper, we study some properties of the indigent module. We give some examples of rings which have an indigent module. We also prove that subinjectivity domain of a module is preserved and reflected under equivalence.

This study was conducted to compare the effect of different ratio of omega-3:omega-6 in total mix ration to blood metabolites and reproductive performance of ewes. A total of 25 young ewes were randomly assigned to five experimental groups and five replications : R0 (without omega-3:omega-6), R1 (omega-3:omega-6 1:8), R2 (omega-3:omega-6 1:6), R3 (omega-3:omega-6 1:3), R4 (omega-3:omega-6 1:2). The parameters measured were feed intake, productive performance, blood glucose and cholesterol pre-mating period. Parameters of characteristic estrous measured were onset of estrous, length of estrous and estrous response. The result showed that ratio of omega-3 and omega-6 1:2 have a greatest daily weight gain and feed efficiency. Ratio of omega-3 and omega-6 1:2 have blood glucose lowest, but highest blood cholesterol. Addition of omega-3 cause to delays onset of estrous. The conclusion is greatest length of estrous, estrous response and pregnacy rate on ratio of omega-3 and omega-6 1:2 in ration.