Linear and multilinear functional identities in a prime ring with applications

This paper approaches some universal-algebraic properties of the two kinds of multilinear functions [Formula: see text] and [Formula: see text] in a prime ring [Formula: see text], where [Formula: see text] are variable elements, [Formula: see text]. We shall demonstrate an algebraic procedure of deriving necessary and sufficient conditions for the two multilinear functional identities [Formula: see text] and [Formula: see text] to hold for all [Formula: see text], [Formula: see text]. Subsequently, we use these multilinear functional identities to describe the invariance properties of the products [Formula: see text] [Formula: see text], [Formula: see text], [Formula: see text] with respect to the eight commonly-used types of generalized inverses of two MP-invertible elements [Formula: see text] and [Formula: see text] in a prime ring [Formula: see text] with an identity element 1 and ∗-involution.

Further results on generalized inverses in rings with involution

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.2958 ◽

2015 ◽

Vol 30 ◽

Cited By ~ 10

Author(s):

N. Castro-Gonzalez ◽

Jianlong Chen ◽

Long Wang

Keyword(s):

Sufficient Conditions ◽

Generalized Inverses ◽

Matrix Product ◽

Explicit Formulas ◽

Block Matrices ◽

Unital Ring ◽

Rings With Involution ◽

Let R be a unital ring with an involution. Necessary and sufficient conditions for the existence of the Bott-Duffin inverse of a in R relative to a pair of self-adjoint idempotents (e, f) are derived. The existence of a {1, 3}-inverse, {1, 4}-inverse, and the Moore-Penrose inverse of a matrix product is characterized, and explicit formulas for their computations are obtained. Some applications to block matrices over a ring are given.

APPLICATION OF INVARIANCE PROPERTIES IN PROBLEMS DEPENDING ON THE PARAMETER

Bulletin Series of Physics & Mathematical Sciences ◽

10.51889/2020-2.1728-7901.06 ◽

2020 ◽

Vol 70 (2) ◽

pp. 47-52

Author(s):

О.D. Apyshev ◽

◽

R.О. Nurkanova ◽

F.S. Аmenova ◽

◽

...

Keyword(s):

Elementary Mathematics ◽

Sufficient Conditions ◽

Solutions To Equations ◽

Systems Of Equations ◽

Invariance Properties ◽

Solution Of Equations ◽

Single Solution ◽

Solving Equations ◽

Invariant Properties

In this paper, we comprehensively consider the problems that depend on the parameter. The necessary and sufficient conditions for the existence of a unique solution of equations and systems of equations that depend on the parameter are investigated. Finding solutions to equations and systems of equations that depend on parameters is one of the most difficult areas of elementary mathematics. To solve them, you need to find ways that require special logical thinking, which must satisfy some additional conditions, for example, to determine a single solution, a solution in a set of all possible values of unknowns, cases when all the solutions of one system are solutions of another, and so on. One of these cases is the use of invariant properties. The article uses various examples of using invariant properties of transformations to consider methods and techniques for solving equations and systems of equations that depend on a single parameter. As a result, due to the presence of invariant properties of transformations, we see that the desired solution is found quickly and easily.

Some additive and multiplicative results for generalized inverses

Filomat ◽

10.2298/fil1509049n ◽

2015 ◽

Vol 29 (9) ◽

pp. 2049-2057

Author(s):

Jovana Nikolov-Radenkovic

Keyword(s):

Hilbert Space ◽

Sufficient Conditions ◽

Generalized Inverses ◽

Reverse Order ◽

Reverse Order Law ◽

Regular Operators

In this paper we give necessary and sufficient conditions for A1{1,3} + A2{1, 3}+ ... + Ak{1,3} ? (A1 + A2 + ... + Ak){1,3} and A1{1,4} + A2{1,4} + ... + Ak{1,4} ? (A1 + A2 + ... + Ak){1,4} for regular operators on Hilbert space. We also consider similar inclusions for {1,2,3}- and {1,2,4}-i inverses. We give some new results concerning the reverse order law for reflexive generalized inverses.

Algebraic Properties of Toeplitz Operators on the Pluriharmonic Bergman Space

Journal of Function Spaces and Applications ◽

10.1155/2013/578436 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12 ◽

Cited By ~ 2

Author(s):

Jingyu Yang ◽

Liu Liu ◽

Yufeng Lu

Keyword(s):

Toeplitz Operator ◽

Bergman Space ◽

Finite Rank ◽

Toeplitz Operators ◽

Sufficient Conditions ◽

Rank Product ◽

Algebraic Properties ◽

Product Problem ◽

Quasihomogeneous Symbols

We study some algebraic properties of Toeplitz operators with radial or quasihomogeneous symbols on the pluriharmonic Bergman space. We first give the necessary and sufficient conditions for the product of two Toeplitz operators with radial symbols to be a Toeplitz operator and discuss the zero-product problem for several Toeplitz operators with radial symbols. Next, we study the finite-rank product problem of several Toeplitz operators with quasihomogeneous symbols. Finally, we also investigate finite rank commutators and semicommutators of two Toeplitz operators with quasihomogeneous symbols.

When Does a Dual Matrix Have a Dual Generalized Inverse?

Symmetry ◽

10.3390/sym13081386 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1386

Author(s):

Firdaus E. Udwadia

Keyword(s):

Explicit Expression ◽

Generalized Inverse ◽

Sufficient Conditions ◽

Generalized Inverses ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Real Matrices ◽

Explicit Expressions

This paper deals with the existence of various types of dual generalized inverses of dual matrices. New and foundational results on the necessary and sufficient conditions for various types of dual generalized inverses to exist are obtained. It is shown that unlike real matrices, dual matrices may not have {1}-dual generalized inverses. A necessary and sufficient condition for a dual matrix to have a {1}-dual generalized inverse is obtained. It is shown that a dual matrix always has a {1}-, {1,3}-, {1,4}-, {1,2,3}-, {1,2,4}-dual generalized inverse if and only if it has a {1}-dual generalized inverse and that every dual matrix has a {2}- and a {2,4}-dual generalized inverse. Explicit expressions, which have not been reported to date in the literature, for all these dual inverses are provided. It is shown that the Moore–Penrose dual generalized inverse of a dual matrix exists if and only if the dual matrix has a {1}-dual generalized inverse; an explicit expression for this dual inverse, when it exists, is obtained irrespective of the rank of its real part. Explicit expressions for the Moore–Penrose dual inverse of a dual matrix, in terms of {1}-dual generalized inverses of products, are also obtained. Several new results related to the determination of dual Moore-Penrose inverses using less restrictive dual inverses are also provided.

Representations of Generalized Inverses and Drazin Inverse of Partitioned Matrix with Banachiewicz-Schur Forms

Mathematical Problems in Engineering ◽

10.1155/2016/9236281 ◽

2016 ◽

Vol 2016 ◽

pp. 1-14 ◽

Cited By ~ 1

Author(s):

Xiaoji Liu ◽

Hongwei Jin ◽

Jelena Višnjić

Keyword(s):

Sufficient Conditions ◽

Schur Complement ◽

Block Matrix ◽

Generalized Inverses ◽

Drazin Inverse ◽

Group Inverse ◽

Generalized Schur Complement ◽

Quotient Property

Representations of 1,2,3-inverses, 1,2,4-inverses, and Drazin inverse of a partitioned matrix M=ABCD related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which 1,2,3-inverses, 1,2,4-inverses, and group inverse of a 2×2 block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of Cvetković-Ilić, 2009, are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement.

On Regular Elements in an Incline

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2010/903063 ◽

2010 ◽

Vol 2010 ◽

pp. 1-12

Author(s):

A. R. Meenakshi ◽

S. Anbalagan

Keyword(s):

Distributive Lattice ◽

Regular Ring ◽

Sufficient Conditions ◽

Generalized Inverses ◽

Regular Elements ◽

Idempotent Semirings ◽

Inclines are additively idempotent semirings in which products are less than (or) equal to either factor. Necessary and sufficient conditions for an element in an incline to be regular are obtained. It is proved that every regular incline is a distributive lattice. The existence of the Moore-Penrose inverse of an element in an incline with involution is discussed. Characterizations of the set of all generalized inverses are presented as a generalization and development of regular elements in a∗-regular ring.

NEUTRAL ELEMENTS, FUNDAMENTAL RELATIONS AND n-ARY HYPERSEMIGROUPS

International Journal of Algebra and Computation ◽

10.1142/s0218196709005226 ◽

2009 ◽

Vol 19 (04) ◽

pp. 567-583 ◽

Cited By ~ 30

Author(s):

B. DAVVAZ ◽

W. A. DUDEK ◽

S. MIRVAKILI

Keyword(s):

Equivalence Relation ◽

Identity Element ◽

Sufficient Conditions ◽

Fundamental Relation ◽

The main tools in the theory of n-ary hyperstructures are the fundamental relations. The fundamental relation on an n-ary hypersemigroup is defined as the smallest equivalence relation so that the quotient would be the n-ary semigroup. In this paper we study neutral elements in n-ary hypersemigroups and introduce a new strongly compatible equivalence relation on an n-ary hypersemigroup so that the quotient is a commutative n-ary semigroup. Also we determine some necessary and sufficient conditions so that this relation is transitive. Finally, we prove that this relation is transitive on an n-ary hypergroup with neutral (identity) element.

The non-orthogonal Cayley–Dickson construction and the octonionic structure of the E8-lattice

Journal of Algebra and Its Applications ◽

10.1142/s0219498817502309 ◽

2017 ◽

Vol 16 (12) ◽

pp. 1750230

Author(s):

Holger P. Petersson

Keyword(s):

Commutative Ring ◽

Linear Form ◽

Sufficient Conditions ◽

Algebraic Properties ◽

Using a conic [Formula: see text] algebra [Formula: see text] over an arbitrary commutative ring, a scalar [Formula: see text] and a linear form [Formula: see text] on [Formula: see text] as input, the non-orthogonal Cayley–Dickson construction produces a conic algebra [Formula: see text] and collapses to the standard (orthogonal) Cayley–Dickson construction for [Formula: see text]. Conditions on [Formula: see text] that are necessary and sufficient for [Formula: see text] to satisfy various algebraic properties (like associativity or alternativity) are derived. Sufficient conditions guaranteeing non-singularity of [Formula: see text] even if [Formula: see text] is singular are also given. As an application, we show how the algebras of Hurwitz quaternions and of Dickson or Coxeter octonions over the rational integers can be obtained from the non-orthogonal Cayley–Dickson construction.

Necessary and Sufficient Conditions for The Solutions of Linear Equation System

Jurnal Matematika Statistika dan Komputasi ◽

10.20956/jmsk.v17i1.10352 ◽

2020 ◽

Vol 17 (1) ◽

pp. 82-88

Author(s):

Gregoria Ariyanti

Keyword(s):

Linear Equation ◽

Commutative Semigroup ◽

Algebraic Structure ◽

Identity Element ◽

Linear Equations ◽

Sufficient Conditions ◽

Equation System ◽

Distributive Property ◽

Linear Equation System ◽

A Semiring is an algebraic structure (S,+,x) such that (S,+) is a commutative Semigroup with identity element 0, (S,x) is a Semigroup with identity element 1, distributive property of multiplication over addition, and multiplication by 0 as an absorbent element in S. A linear equations system over a Semiring S is a pair (A,b) where A is a matrix with entries in S and b is a vector over S. This paper will be described as necessary or sufficient conditions of the solution of linear equations system over Semiring S viewed by matrix X that satisfies AXA=A, with A in S. For a matrix X that satisfies AXA=A, a linear equations system Ax=b has solution x=Xb+(I-XA)h with arbitrary h in S if and only if AXb=b.