scholarly journals ON PRIME-LIKE RADICALS

2010 ◽  
Vol 82 (1) ◽  
pp. 113-119 ◽  
Author(s):  
S. TUMURBAT ◽  
H. FRANCE-JACKSON
Keyword(s):  
Polynomial Ring ◽  
Prime Ring ◽  
Open Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Prime Radical ◽  
Polynomial Rings ◽  
Necessary And Sufficient

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.

Automorphisms and Derivations of Skew Polynomial Rings

1992 ◽  
Vol 35 (1) ◽  
pp. 126-132 ◽  
Author(s):  
Mary P. Rosen ◽  
Jerry D. Rosen
Keyword(s):  
Polynomial Ring ◽  
Prime Ring ◽  
Sufficient Conditions ◽  
Necessary And 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].

Jacobson Radicals of Skew Polynomial Rings of Derivation Type

2014 ◽  
Vol 57 (3) ◽  
pp. 609-613 ◽  
Author(s):  
Alireza Nasr-Isfahani
Keyword(s):  
Polynomial Ring ◽  
Jacobson Radical ◽  
Sufficient Conditions ◽  
Necessary And 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.

scholarly journals MAXIMAL COMMUTATIVE SUBRINGS AND SIMPLICITY OF ORE EXTENSIONS

2013 ◽  
Vol 12 (04) ◽  
pp. 1250192 ◽  
Author(s):  
JOHAN ÖINERT ◽  
JOHAN RICHTER ◽  
SERGEI D. SILVESTROV
Keyword(s):  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Differential Polynomial ◽  
Nonzero Ideal ◽  
Polynomial Rings ◽  
Ore Extension ◽  
Differential Polynomial Ring ◽  
Ore Extensions ◽  
Necessary And Sufficient ◽  
Commutative Subring

The aim of this paper is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x; id R, δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x; σ, δ] is a maximal commutative subring containing R and, in the case when σ = id R, we show that it intersects every nonzero ideal of R[x; id R, δ] nontrivially. Using this we show that if R is δ-simple and maximal commutative in R[x; id R, δ], then R[x; id R, δ] is simple. We also show that under some conditions on R the converse holds.

scholarly journals Pairs of rings with a bijective correspondence between the prime spectra

1993 ◽  
Vol 55 (2) ◽  
pp. 238-245
Author(s):  
E. Jespers ◽  
P. Wauters
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Free Abelian Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Rings ◽  
Prime Spectrum ◽  
Bijective Correspondence ◽  
Natural Mapping ◽  
Necessary And Sufficient

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.

scholarly journals Factorizations of Invertible Symmetric Matrices over Polynomial Rings with Involution

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Rings ◽  
Symmetric Matrices ◽  
Matrix Polynomials ◽  
Elementary Divisors ◽  
Rings With Involution ◽  
Ring Of Polynomials ◽  
Necessary And Sufficient

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.

scholarly journals Some Extensions of Generalized Morphic Rings and EM-rings

2018 ◽  
Vol 26 (1) ◽  
pp. 111-123
Author(s):  
Manal Ghanem ◽  
Emad Abu Osba
Keyword(s):  
Commutative Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Rings ◽  
Necessary And Sufficient

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

scholarly journals A Note on Isomorphism Theorems for Semigroups of Order-Preserving Transformations with Restricted Range

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Phichet Jitjankarn ◽  
Thitarie Rungratgasame
Keyword(s):  
Open Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Restricted Range ◽  
Infinite Domain ◽  
Necessary And Sufficient ◽  
Proof Strategy ◽  
Isomorphism Theorems

Finding necessary and sufficient conditions for isomorphism between two semigroups of order-preserving transformations over an infinite domain with restricted range was an open problem. In this paper, we show a proof strategy to answer that question.

scholarly journals Subnormality of Powers of Multivariable Weighted Shifts

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Sang Hoon Lee ◽  
Woo Young Lee ◽  
Jasang Yoon
Keyword(s):  
Hilbert Space ◽  
Open Problem ◽  
Operator Theory ◽  
Sufficient Conditions ◽  
Weighted Shift ◽  
Necessary And Sufficient Conditions ◽  
Weighted Shifts ◽  
Hilbert Space Operators ◽  
Necessary And Sufficient

Given a pair T ≡ T 1 , T 2 of commuting subnormal Hilbert space operators, the Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for the existence of a commuting pair N ≡ N 1 , N 2 of normal extensions of T 1 and T 2 ; in other words, T is a subnormal pair. The LPCS is a longstanding open problem in the operator theory. In this paper, we consider the LPCS of a class of powers of 2 -variable weighted shifts. Our main theorem states that if a “corner” of a 2-variable weighted shift T = W α , β ≔ T 1 , T 2 is subnormal, then T is subnormal if and only if a power T m , n ≔ T 1 m , T 2 n is subnormal for some m , n ≥ 1 . As a corollary, we have that if T is a 2-variable weighted shift having a tensor core or a diagonal core, then T is subnormal if and only if a power of T is subnormal.

scholarly journals Necessary and Sufficient Conditions for Oscillation of Solutions to Second-Order Neutral Differential Equations with Impulses

2020 ◽  
Vol 76 (1) ◽  
pp. 157-170 ◽  
Author(s):  
Shyam Sundar Santra
Keyword(s):  
Open Problem ◽  
Convergence Theorem ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Neutral Differential Equations ◽  
Dominated Convergence ◽  
Necessary And Sufficient ◽  
Oscillation Of Solutions

AbstractIn this work, necessary and sufficient conditions for oscillation of solutions of second-order neutral impulsive differential system\left\{ {\matrix{{{{\left( {r\left( t \right){{\left( {z'\left( t \right)} \right)}^\gamma }} \right)}^\prime } + q\left( t \right){x^\alpha }\left( {\sigma \left( t \right)} \right) = 0,} \hfill & {t \ge {t_0},\,\,\,t \ne {\lambda _k},} \hfill \cr {\Delta \left( {r\left( {{\lambda _k}} \right){{\left( {z'\left( {{\lambda _k}} \right)} \right)}^\gamma }} \right) + h\left( {{\lambda _k}} \right){x^\alpha }\left( {\sigma \left( {{\lambda _k}} \right)} \right) = 0,} \hfill & {k \in \mathbb{N}} \hfill \cr } } \right. are established, where z\left( t \right) = x\left( t \right) + p\left( t \right)x\left( {\tau \left( t \right)} \right)Under the assumption \int {^\infty {{\left( {r\left( \eta \right)} \right)}^{ - 1/\alpha }}d\eta = \infty } two cases when γ>α and γ<α are considered. The main tool is Lebesgue’s Dominated Convergence theorem. Examples are given to illustrate the main results, and state an open problem.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj&gt; 0 for eachj&gt; 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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