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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.

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 [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 [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.

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.

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.

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.

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

Let E and F be Banach lattice with F is a Dedekind ? -complete. We characterize the generalized classes ( X ( E ) , Y ( F ) ) of all infinite matrices of regular operators from E to F, for X ? { ? , c , ? ∞ } and Y ? { c , c 0 , ? ∞ } .

There are various approaches to transfer the prime ideal notion to prime submodule. The denition of primeness in modules have appeared in many ways. In this paper we dene endo-prime submodules and show that this denition and prime submodules introduced by Sanh et al. [8] are coincide. Moreover, we also present the endo-prime radical and its properties. Then we apply our results to the endo-multiplication modules, i.e. a multiplication module over its ring endomorphisms. Mathematics Subject Classication: 16D50, 16D70, 16D80

Let [Formula: see text] be a positively graded ring which is a sub-ring of strongly graded ring of type [Formula: see text], where [Formula: see text] is a Noetherian prime ring. We define a concept of [Formula: see text]-invariant maximal order and show that [Formula: see text] is a maximal order if and only if [Formula: see text] is a [Formula: see text]-invariant maximal order. If [Formula: see text] is a maximal order, then we completely describe all [Formula: see text]-invertible ideals. As an application, we show that [Formula: see text] is a generalized Dedekind prime ring if and only if [Formula: see text] is a [Formula: see text]-invariant generalized Dedekind prime ring. We give examples of [Formula: see text]-invariant generalized Dedekind prime rings but neither generalized Dedekind prime rings nor maximal orders.

Let R and S be arbitrary rings. In the algebraic structure it is known that the R-module structure is a generalization of a vector space. As in the ring structure, in the R-module some previous researchers have defined R-module homomorphisms, the types of R-module homomorphisms, the properties of R-module homomorphisms, and the fundamental theorem of R-module isomorphisms. On the other hand, the R-module structure has been generalized to the (R, S)-module structure. However, research and discussion related to (R, S)-modules are still a bit worked out. Therefore, in this paper we present the definition of (R, S)-module homomorphisms, the types of (R, S)-module homomorphisms, the properties of (R, S)-module homomorphisms, and the fundamental theorem of (R, S)-module isomorphisms.

In this paper, the space of a Banach lattice valued sequences is defined. Further, we will construct sufficient and necessary conditions for the boundedness of the superposition operators on the space.

Prime essential ring and generalized prime essential rings can be used to determine whether a supernilpotent radical is a special radical. Special radical is one of the most important structure in the development of radical theory of rings. Since prime essential rings have an important role in radical theory of rings, we describe some properties of prime essential rings as well as their generalization. We show that every prime essential rings is a subdirectly reducible ring. Furthermore, the prime heart of prime essential ring is zero, so is the semiprime heart of prime an essential ring.

The primeness of submodules in a module M could be identified by using the fraction of module. On his paper, Jabbar defined an S(N)-locally prime submodule by applying this idea. In this work, we conduct some further properties and a weaker condition of S(N)-locally prime submodules. It is already well known that the dual of prime submodules is second submodules. Hence we also introduce a possibility to dualize the notion of S(N)-locally prime submodules into S(0)-locally second submodules. We investigate the characterizations of locally S(0)-second submodules and give some conditions of rings such that all modules over it will have S(0)-locally second submodules.

Let μ be a special class of rings, i.e. class of prime associative rings which is closed under ideals and essential extensions. Let *μ be the class of all *μ–rings, that is, rings in μ such that the factor ring is not in μ for every nonzero ideal of R. Apart from their interesting properties, *μ–rings from a generalization of the important class of *–rings introduced and studied in [4], [7], [8], and [14]. France-Jackson and Roux proved useful in describing minimal elements of lattices of hereditary radicals [9].It was observed in [9] that some special classes μ of rings, such as the class of all simple rings with unity, for example, coincide with *μ and the question was asked to describe all such special classes. We will give a partial answer to this question by noting that, in fact, this holds for every special class of simple rings with unity. However, the converse is not true.

We recall the notion of prie modules and use the analogue technique to define prime comodules and corings. Moreover, the related properties are of interest. We investigate the relation of primeness of C-comodule M and the dual algebra *C of a coring C, the relation to projectivity of a coring in the associated category, the implication of the primeness to the injective hull and product of prime coalgebras. DOI : http://dx.doi.org/10.22342/jims.14.1.58.13-24

Many observations about coalgebras were inspired by comparable situations for algebras. Despite the prominent role of prime algebras, the theory of a corresponding notion for coalgebras was not well understood so far. Coalgebras C over fields may be called coprime provided the dual algebra C* is prime. This definition, however, is not intrinsic—it strongly depends on the base ring being a field. The purpose of the article is to provide a better understanding of related notions for coalgebras over commutative rings by employing traditional methods from (co)module theory, in particular (pre)torsion theory. Dualizing classical primeness condition, coprimeness can be defined for modules and algebras. These notions are developed for modules and then applied to comodules. We consider prime and coprime, fully prime and fully coprime, strongly prime and strongly coprime modules and comodules. In particular, we obtain various characterisations of prime and coprime coalgebras over rings and fields.