Fuzzy Small Submodule and Jacobson -Radical

Direct Sum ◽

Necessary And Sufficient ◽

Quotient Modules ◽

Small Submodule

Using the notion of fuzzy small submodules of a module, we introduce the concept of fuzzy coessential extension of a fuzzy submodule of a module. We attempt to investigate various properties of fuzzy small submodules of a module. A necessary and sufficient condition for fuzzy small submodules is established. We investigate the nature of fuzzy small submodules of a module under fuzzy direct sum. Fuzzy small submodules of a module are characterized in terms of fuzzy quotient modules. This characterization gives rise to some results on fuzzy coessential extensions. Finally, a relation between small -submodules and Jacobson -radical is established.

Chaos for Cosine Operator Functions on Groups

Abstract and Applied Analysis ◽

10.1155/2014/603234 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Chung-Chuan Chen

Keyword(s):

Locally Compact Group ◽

Sufficient Condition ◽

Locally Compact ◽

Direct Sum ◽

Cosine Operator ◽

Weight Condition ◽

Operator Functions ◽

Cosine Operator Functions ◽

Let1≤p<∞andGbe a locally compact group. We characterize chaotic cosine operator functions, generated by weighted translations on the Lebesgue spaceLp(G), in terms of the weight condition. In particular, chaotic cosine operator functions and chaotic weighted translations can only occur simultaneously. We also give a necessary and sufficient condition for the direct sum of a sequence of cosine operator functions to be chaotic.

Direct-sum decomposition of atomic and orthogonally complete rings

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700006777 ◽

1970 ◽

Vol 11 (3) ◽

pp. 357-361 ◽

Cited By ~ 5

Author(s):

Alexander Abian

Keyword(s):

Natural Number ◽

Sufficient Condition ◽

Direct Sum ◽

Direct Sum Decomposition ◽

Image Position ◽

In this paper we give a necessary and sufficient condition for decomposition (as a direct sum of fields) of a ring R in which for every x ∈ R there exists a (and hence the smallest) natural number n(x) > 1 such that . We would like to emphasize that in what follows R stands for a ring every element x of which satisfies (1).

Sifat-Sifat Representasi Indekomposabel The Properties of Indecomposable Representations

Natural Science Journal of Science and Technology ◽

10.22487/25411969.2019.v8.i3.14598 ◽

2019 ◽

Vol 8 (3) ◽

Author(s):

Vika Yugi Kurniawan

Keyword(s):

Vector Space ◽

Directed Graph ◽

Linear Mapping ◽

Sufficient Condition ◽

Paper Study ◽

Direct Sum ◽

Indecomposable Representations ◽

Finite Set ◽

A directed graph is also called as a quiver where is a finite set of vertices, is a set of arrows, and are two maps from to . A representation of a quiver is an assignment of a vector space to each vertex of and a linear mapping to each arrow. We denote by the direct sum of representasions and of a quiver . A representation is called indecomposable if is not ishomorphic to a direct sum of non-zero representations. This paper study about the properties of indecomposable representations. These properties will be used to investigate the necessary and sufficient condition of indecomposable representations.

On Properties of ClassA(n)andn-Paranormal Operators

Abstract and Applied Analysis ◽

10.1155/2014/629061 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5

Author(s):

Xiaochun Li ◽

Fugen Gao

Keyword(s):

Positive Integer ◽

Tensor Products ◽

Unit Vector ◽

Sufficient Condition ◽

Direct Sum ◽

Paranormal Operator ◽

Letnbe a positive integer, and an operatorT∈B(ℋ)is called a classA(n)operator ifT1+n2/1+n≥|T|2andn-paranormal operator ifT1+nx1/1+n≥||Tx||for every unit vectorx∈ℋ, which are common generalizations of classAand paranormal, respectively. In this paper, firstly we consider the tensor products for classA(n)operators, giving a necessary and sufficient condition forT⊗Sto be a classA(n)operator whenTandSare both non-zero operators; secondly we consider the properties forn-paranormal operators, showing that an-paranormal contraction is the direct sum of a unitary and aC.0completely non-unitary contraction.

Derivations on some (possibly non-separable) C*-algebras

Glasgow Mathematical Journal ◽

10.1017/s0017089500004456 ◽

1981 ◽

Vol 22 (1) ◽

pp. 43-56 ◽

Cited By ~ 2

Author(s):

J. P. Sproston

Keyword(s):

Sufficient Condition ◽

Finite Degree ◽

C Algebras ◽

Direct Sum ◽

In an important recent paper [4], G. A. Elliott has given a necessary and sufficient condition for every derivation on a separable C*-algebra with identity to be inner. Indeed, Elliott's condition has since been shown, by Akemann and Pedersen, to be equivalent to the C*-algebra being a finite direct sum of C*-algebras which are either homogeneous of finite degree or simple [8, Corollary 3.10].

On the Yoneda Ext-Algebras of Semiperfect Algebras

Algebra Colloquium ◽

10.1142/s1005386708000217 ◽

2008 ◽

Vol 15 (02) ◽

pp. 207-222 ◽

Cited By ~ 5

Author(s):

Jiwei He ◽

Yu Ye

Keyword(s):

Jacobson Radical ◽

Sufficient Condition ◽

Yoneda Algebra ◽

Ideal Formula ◽

Morita Equivalent ◽

Morita Invariant ◽

The Right ◽

It is proved that the Yoneda Ext-algebras of Morita equivalent semiperfect algebras are graded equivalent. The Yoneda Ext-algebras of noetherian semiperfect algebras are studied in detail. Let A be a noetherian semiperfect algebra with Jacobson radical J. We construct a right ideal [Formula: see text] of the Yoneda algebra [Formula: see text], which plays an important role in the discussion of the structure of E(A). An extra grading is introduced to [Formula: see text], by which we give a description of the right ideal of E(A) generated by [Formula: see text], and we give a necessary and sufficient condition for a notherian semiperfect algebra to be higher quasi-Koszul. Finally, it is shown that the quasi-Koszulity of a noetherian semiperfect algebra is a Morita invariant.

CONDITION OF FINITENESS OF COLENGTH OF VARIETY OF LEIBNITZ ALGEBRAS

Vestnik of Samara University Natural Science Series ◽

10.18287/2541-7525-2014-20-10-84-90 ◽

2017 ◽

Vol 20 (10) ◽

pp. 84-90

Author(s):

A.V. Polovinkina ◽

T.V. Skoraya

Keyword(s):

Symmetric Group ◽

Free Algebra ◽

Sufficient Condition ◽

Base Field ◽

Direct Sum ◽

Zero Characteristic ◽

Necessary And Sufficient ◽

Relationship Of ◽

The Relationship

This paper is devoted to the varieties of Leibnitz algebras over a ﬁeld of zero characteristic. All information about the variety in case of zero characteristic of the base ﬁeld is contained in the space of multilinear elements of its relatively free algebra. Multilinear component of variety is considered as a module of symmetric group and splits into a direct sum of irreducible submodules, the sum of multiplicities of which is called colength of variety. This paper investigates the identities that are performed in varieties with ﬁnite colength and also the relationship of this varieties with known varieties of Lie and Leibnitz algebras with this property. We prove necessary and suﬃcient condition for a ﬁniteness of colength of variety of Leibnitz algebras.

On split regular BiHom-Poisson color algebras

Open Mathematics ◽

10.1515/math-2021-0039 ◽

2021 ◽

Vol 19 (1) ◽

pp. 600-613

Author(s):

Yaling Tao ◽

Yan Cao

Keyword(s):

Natural Extension ◽

Sufficient Condition ◽

Poisson Algebras ◽

Direct Sum ◽

The Family ◽

Abstract The purpose of this paper is to introduce the class of split regular BiHom-Poisson color algebras, which can be considered as the natural extension of split regular BiHom-Poisson algebras and of split regular Poisson color algebras. Using the property of connections of roots for this kind of algebras, we prove that such a split regular BiHom-Poisson color algebra L L is of the form L = ⊕ [ α ] ∈ Λ / ∼ I [ α ] L={\oplus }_{\left[\alpha ]\in \Lambda \text{/} \sim }{I}_{\left[\alpha ]} with I [ α ] {I}_{\left[\alpha ]} a well described (graded) ideal of L L , satisfying [ I [ α ] , I [ β ] ] + I [ α ] I [ β ] = 0 \left[{I}_{\left[\alpha ]},{I}_{\left[\beta ]}]+{I}_{\left[\alpha ]}{I}_{\left[\beta ]}=0 if [ α ] ≠ [ β ] \left[\alpha ]\ne \left[\beta ] . In particular, a necessary and sufficient condition for the simplicity of this algebra is determined, and it is shown that L L is the direct sum of the family of its simple (graded) ideals.