Factorizations of Invertible Symmetric Matrices over Polynomial Rings with Involution

Polynomial Rings ◽

Symmetric Matrices ◽

Matrix Polynomials ◽

Elementary Divisors ◽

Rings With Involution ◽

Ring Of Polynomials ◽

By means of the notions of infinite elementary divisors, dual and generalized dual matrix polynomials, we find necessary and sufficient conditions for the existence of factorizations of invertible symmetric matrices over ring of polynomials with involution.

Jacobson Radicals of Skew Polynomial Rings of Derivation Type

Canadian Mathematical Bulletin ◽

10.4153/cmb-2014-018-9 ◽

2014 ◽

Vol 57 (3) ◽

pp. 609-613 ◽

Cited By ~ 3

Author(s):

Alireza Nasr-Isfahani

Keyword(s):

Polynomial Ring ◽

Jacobson Radical ◽

Sufficient Conditions ◽

Polynomial Rings ◽

Skew Polynomial Ring ◽

Skew Polynomial Rings ◽

Base Ring ◽

Necessary And Sufficient ◽

Skew Polynomial

AbstractWe provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type.

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.

k-Kernel Symmetric Matrices

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2009/926217 ◽

2009 ◽

Vol 2009 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

A. R. Meenakshi ◽

D. Jaya Shree

Keyword(s):

Scalar Product ◽

Sufficient Conditions ◽

Symmetric Matrices ◽

Fuzzy Matrix ◽

Basic Properties ◽

In this paper we present equivalent characterizations ofk-Kernel symmetric Matrices. Necessary and sufficient conditions are determined for a matrix to bek-Kernel Symmetric. We give some basic results of kernel symmetric matrices. It is shown that k-symmetric impliesk-Kernel symmetric but the converse need not be true. We derive some basic properties ofk-Kernel symmetric fuzzy matrices. We obtain k-similar and scalar product of a fuzzy matrix.

Weighted Reverse Order Law for the Weighted Moore-Penrose Inverse in Rings with Involution

Annals of the Alexandru Ioan Cuza University - Mathematics ◽

10.2478/aicu-2014-0041 ◽

2014 ◽

Vol 0 (0) ◽

Author(s):

Dijana Mosić ◽

Dragan S. Djordjević

Keyword(s):

Sufficient Conditions ◽

Reverse Order ◽

Reverse Order Law ◽

Rings With Involution ◽

Abstract We investigate some necessary and sufficient conditions for the reverse order law for the weighted Moore-Penrose inverse in rings with involution.

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

Pairs of rings with a bijective correspondence between the prime spectra

10.1017/s1446788700032055 ◽

1993 ◽

Vol 55 (2) ◽

pp. 238-245

Author(s):

E. Jespers ◽

P. Wauters

Keyword(s):

Abelian Group ◽

Commutative Ring ◽

Free Abelian Group ◽

Sufficient Conditions ◽

Polynomial Rings ◽

Prime Spectrum ◽

Bijective Correspondence ◽

Natural Mapping ◽

AbstractLet A be a subring of a commutative ring B. If the natural mapping from the prime spectrum of B to the prime spectrum of A is injective (respectively bijective) then the pair (A, B) is said to have the injective (respectively bijective) Spec-map. We give necessary and sufficient conditions for a pair of rings A and B graded by a free abelian group to have the injective (respectively bijective) Spec-map. For this we first deal with the polynomial case. Let l be a field and k a subfield. Then the pair of polynomial rings (k[X], l[X]) has the injective Spec-map if and only if l is a purely inseparable extension of k.

Standard pairs and existence of symmetric multiscaling functions

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691319500577 ◽

2019 ◽

Vol 18 (02) ◽

pp. 1950057

Author(s):

A. T. Mithun ◽

M. C. Lineesh

Keyword(s):

Matrix Polynomial ◽

Sufficient Conditions ◽

Standard Pair ◽

Compactly Supported ◽

Matrix Polynomials ◽

The Matrix ◽

Symbol Function ◽

Necessary And Sufficient ◽

Multiscaling Function

Construction of multiwavelets begins with finding a solution to the multiscaling equation. The solution is known as multiscaling function. Then, a multiwavelet basis is constructed from the multiscaling function. Symmetric multiscaling functions make the wavelet basis symmetric. The existence and properties of the multiscaling function depend on the symbol function. Symbol functions are trigonometric matrix polynomials. A trigonometric matrix polynomial can be constructed from a pair of matrices known as the standard pair. The square matrix in the pair and the matrix polynomial have the same spectrum. Our objective is to find necessary and sufficient conditions on standard pairs for the existence of compactly supported, symmetric multiscaling functions. First, necessary as well as sufficient conditions on the standard pairs for the existence of symbol functions corresponding to compactly supported multiscaling functions are found. Then, the necessary and sufficient conditions on the class of standard pairs, which make the multiscaling function symmetric, are derived. A method to construct symbol function corresponding to a compactly supported, symmetric multiscaling function from an appropriate standard pair is developed.

Some Extensions of Generalized Morphic Rings and EM-rings

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2018-0007 ◽

2018 ◽

Vol 26 (1) ◽

pp. 111-123

Author(s):

Manal Ghanem ◽

Emad Abu Osba

Keyword(s):

Commutative Ring ◽

Sufficient Conditions ◽

Polynomial Rings ◽

AbstractLet R be a commutative ring with unity. The main objective of this article is to study the relationships between PP-rings, generalized morphic rings and EM-rings. Although PP-rings are included in the later rings, the converse is not in general true. We put necessary and sufficient conditions to ensure the converse using idealization and polynomial rings

Automorphisms and Derivations of Skew Polynomial Rings

Canadian Mathematical Bulletin ◽

10.4153/cmb-1992-018-6 ◽

1992 ◽

Vol 35 (1) ◽

pp. 126-132 ◽

Cited By ~ 3

Author(s):

Mary P. Rosen ◽

Jerry D. Rosen

Keyword(s):

Polynomial Ring ◽

Prime Ring ◽

Sufficient Conditions ◽

Polynomial Rings ◽

Skew Polynomial Ring ◽

Skew Polynomial Rings ◽

Inner Automorphisms ◽

Necessary And Sufficient ◽

Skew Polynomial

AbstractFor a prime ring R and σ ∊ Aut(R), we determine the group of Rstabilizing automorphisms of the skew polynomial ring R[x; σ]. In the case where R is simple, we characterize the X-inner automorphisms of R[x; σ]. We also provide necessary and sufficient conditions for a σ -commuting derivation of a prime ring R to extend to a derivation of R[x; σ].

ON PRIME-LIKE RADICALS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972709001129 ◽

2010 ◽

Vol 82 (1) ◽

pp. 113-119 ◽

Cited By ~ 3

Author(s):

S. TUMURBAT ◽

H. FRANCE-JACKSON

Keyword(s):

Polynomial Ring ◽

Prime Ring ◽

Open Problem ◽

Sufficient Conditions ◽

Prime Radical ◽

Polynomial Rings ◽

AbstractA radical γ is prime-like if, for every prime ring A, the polynomial ring A[x] is γ-semisimple. In this paper, we study properties of prime-like radicals. In particular, we give necessary and sufficient conditions for a radical γ containing the prime radical β to be prime-like. This allows us to easily find distinct special radicals that coincide on simple rings and on polynomial rings, which answers a question put by Ferrero. It also allows us to reformulate a long-standing open problem of Gardner in terms of prime-like radicals.

The Moore-Penrose inverse in rings with involution

Filomat ◽

10.2298/fil1918791x ◽

2019 ◽

Vol 33 (18) ◽

pp. 5791-5802 ◽

Cited By ~ 1

Author(s):

Sanzhang Xu ◽

Jianlong Chen

Keyword(s):

Sufficient Conditions ◽

Unital Ring ◽

Rings With Involution ◽

Let R be a unital ring with involution. In this paper, we first show that for an element a 2 R, a is Moore-Penrose invertible if and only if a is well-supported if and only if a is co-supported. Moreover, several new necessary and sufficient conditions for the existence of the Moore-Penrose inverse of an element in a ring R are obtained. In addition, the formulae of the Moore-Penrose inverse of an element in a ring are presented.