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4352. Commutativity of rings through a Streb’s result
- Creator:
- Khan, Moharram A.
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- commutators, division rings, factorsubrings, polynomial identities, and torsion-free rings
- Language:
- English
- Description:
- In this paper we investigate commutativity of rings with unity satisfying any one of the properties: \[ \begin{aligned} &\lbrace 1- g(yx^{m}) \rbrace \ [yx^{m} - x^{r} f (yx^{m}) \ x^s, x] \lbrace 1- h (yx^{m}) \rbrace = 0, \\&\lbrace 1- g(yx^{m}) \rbrace \ [x^{m} y - x^{r} f (yx^{m}) x^{s}, x] \lbrace 1- h (yx^{m}) \rbrace = 0, \\&y^{t} [x,y^{n}] = g (x) [f (x), y] h (x)\ {\mathrm and} \ \ [x,y^{n}] \ y^{t} = g (x) [f (x), y] h (x) \end{aligned} \] for some $f(X)$ in $X^{2} {\mathbb Z}[X]$ and $g(X)$, $ h(X)$ in ${\mathbb Z} [X]$, where $m \ge 0$, $ r \ge 0$, $ s \ge 0$, $ n > 0$, $ t > 0$ are non-negative integers. We also extend these results to the case when integral exponents in the underlying conditions are no longer fixed, rather they depend on the pair of ring elements $x$ and $y$ for their values. Further, under different appropriate constraints on commutators, commutativity of rings has been studied. These results generalize a number of commutativity theorems established recently.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4353. Commutativity of rings with constraints involving a subset
- Creator:
- Khan, Moharram A.
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- commutativity theorems, Jacobson radicals, nilpotent elements, periodic rings, and torsion-free rings
- Language:
- English
- Description:
- Suppose that $R$ is an associative ring with identity $1$, $J(R)$ the Jacobson radical of $R$, and $N(R)$ the set of nilpotent elements of $R$. Let $m \ge 1$ be a fixed positive integer and $R$ an $m$-torsion-free ring with identity $1$. The main result of the present paper asserts that $R$ is commutative if $R$ satisfies both the conditions (i) $[x^m,y^m] = 0$ for all $x,y \in R \setminus J(R)$ and (ii) $[(xy)^m + y^mx^m, x] = 0 = [(yx)^m + x^my^m, x]$, for all $x,y \in R \setminus J(R)$. This result is also valid if (i) and (ii) are replaced by (i)$^{\prime }$ $[x^m,y^m] = 0$ for all $x,y \in R \setminus N(R)$ and (ii)$^{\prime }$ $[(xy)^m + y^m x^m, x] = 0 = [(yx)^m + x^m y^m, x]$ for all $x,y \in R\backslash N(R) $. Other similar commutativity theorems are also discussed.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4354. Commutativity of rings with polynomial constraints
- Creator:
- Khan, Moharram A.
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- automorphism, commutativity, local ring, polynomial identity, and $s$-unital ring
- Language:
- English
- Description:
- Let $p$, $ q$ and $r$ be fixed non-negative integers. In this note, it is shown that if $R$ is left (right) $s$-unital ring satisfying $[f(x^py^q) - x^ry, x] = 0$ ($[f(x^py^q) - yx^r, x] = 0$, respectively) where $f(\lambda ) \in {\lambda }^2{\mathbb Z}[\lambda ]$, then $R$ is commutative. Moreover, commutativity of $R$ is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4355. Commutator subgroups of the extended Hecke groups $\bar{H}(\lambda_q)$.
- Creator:
- Şahin, Recep , Bizim, Osman , and Cangul, I. N.
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Hecke group, extended Hecke group, and commutator subgroup
- Language:
- English
- Description:
- Hecke groups $H(\lambda _q)$ are the discrete subgroups of ${\mathrm PSL}(2,\mathbb{R})$ generated by $S(z)=-(z+\lambda _q)^{-1}$ and $T(z)=-\frac{1}{z} $. The commutator subgroup of $H$($\lambda _q)$, denoted by $H^{\prime }(\lambda _q)$, is studied in [2]. It was shown that $H^{\prime }(\lambda _q)$ is a free group of rank $q-1$. Here the extended Hecke groups $\bar{H}(\lambda _q)$, obtained by adjoining $R_1(z)=1/\bar{z}$ to the generators of $H(\lambda _q)$, are considered. The commutator subgroup of $\bar{H}(\lambda _q)$ is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the $H(\lambda _q)$ case, the index of $H^{\prime }(\lambda _q)$ is changed by $q$, in the case of $\bar{H}(\lambda _q)$, this number is either 4 for $q$ odd or 8 for $q$ even.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4356. Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent
- Creator:
- Wang, Hongbin
- Format:
- print, bez média, and svazek
- Type:
- model:article and TEXT
- Subject:
- matematika, integrály, mathematics, integrals, Herz space, variable exponent, commutator, Marcinkiewicz integral, 13, and 51
- Language:
- English
- Description:
- Let \Omega \in L^{s}\left ( S^{n-1} \right ) for s\geqslant 1 be a homogeneous function of degree zero and b a BMO function. The commutator generated by the Marcinkiewicz integral μΩ and b is defined by \left[ {b,{\mu _\Omega }} \right](f)(x) = {\left( {\int_0^\infty {{{\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega (x - y)}}{{{{\left| {x - y} \right|}^{n - 1}}}}\left[ {b(x) - b(y)} \right]f(y){\text{d}}y} } \right|}^2}\frac{{{\text{d}}t}}{{{t^3}}}} } \right)^{1/2}}. In this paper, the author proves the \left (L^{p\left ( \cdot \right )}\left ( \mathbb{R}^{n} \right ),L^{p\left ( \cdot \right )}\left ( \mathbb{R}^{n} \right ) \right )-boundedness of the Marcinkiewicz integral operator μΩ and its commutator [b, μΩ ] when p(·) satisfies some conditions. Moreover, the author obtains the corresponding result about μΩ and [b, μΩ ] on Herz spaces with variable exponent., Hongbin Wang., and Obsahuje seznam literatury
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4357. Commutators of singular integrals on spaces of homogeneous type
- Creator:
- Pradolini, Gladis and Salinas, Oscar
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- commutators, spaces of homogeneous type, and weights
- Language:
- English
- Description:
- In this work we prove some sharp weighted inequalities on spaces of homogeneous type for the higher order commutators of singular integrals introduced by R. Coifman, R. Rochberg and G. Weiss in Factorization theorems for Hardy spaces in several variables, Ann. Math. 103 (1976), 611–635. As a corollary, we obtain that these operators are bounded on $L^{p}(w)$ when $w$ belongs to the Muckenhoupt’s class $A_{p}$, $p>1$. In addition, as an important tool in order to get our main result, we prove a weighted Fefferman-Stein type inequality on spaces of homogeneous type, which we have not found previously in the literature.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4358. Commutators of sublinear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces
- Creator:
- Guliyev, Vagif Sabir, Karaman, Turhan, Mustafayev, Rza Chingiz, and Şerbetçi, Ayhan
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- generalized weighted Morrey space, sublinear operator;, commutator, BMO space, maximal operator, and Calderón-Zygmund operator
- Language:
- English
- Description:
- In this paper, the boundedness of a large class of sublinear commutator operators $T_{b}$ generated by a Calderón-Zygmund type operator on a generalized weighted Morrey spaces $M_{p,\varphi }(w)$ with the weight function $w$ belonging to Muckenhoupt's class $A_{p}$ is studied. When $1<p<\infty $ and $b \in {\rm BMO}$, sufficient conditions on the pair $(\varphi _1,\varphi _2)$ which ensure the boundedness of the operator $T_{b}$ from $M_{p,\varphi _1}(w)$ to $M_{p,\varphi _2}(w)$ are found. In all cases the conditions for the boundedness of $T_{b}$ are given in terms of Zygmund-type integral inequalities on $(\varphi _1,\varphi _2)$, which do not require any assumption on monotonicity of $\varphi _1(x,r)$, $\varphi _2(x,r)$ in $r$. Then these results are applied to several particular operators such as the pseudo-differential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4359. Commutators of the fractional maximal function on variable exponent Lebesgue spaces
- Creator:
- Zhang, Pu and Wu, Jianglong
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- commutator, BMO, fractional maximal function, and variable exponent Lebesgue space
- Language:
- English
- Description:
- Let $M_{\beta }$ be the fractional maximal function. The commutator generated by $M_{\beta }$ and a suitable function $b$ is defined by $[M_{\beta },b]f = M_{\beta }(bf)-bM_{\beta }(f)$. Denote by $\mathscr {P}(\mathbb R^{n})$ the set of all measurable functions $p(\cdot )\colon \mathbb R^{n}\to [1,\infty )$ such that $$ 1< p_{-}:=\mathop {\rm ess inf}_{x\in \mathbb R^n}p(x) \quad \text {and}\quad p_{+}:=\mathop {\rm ess sup}_{x\in \mathbb R^n}p(x)<\infty , $$ and by $\mathscr {B}(\mathbb R^{n})$ the set of all $p(\cdot ) \in \mathscr {P}(\mathbb R^{n})$ such that the Hardy-Littlewood maximal function $M$ is bounded on $L^{p(\cdot )}(\mathbb R^{n})$. In this paper, the authors give some characterizations of $b$ for which $[M_{\beta },b]$ is bounded from $L^{p(\cdot )}(\mathbb R ^{n})$ into $L^{q(\cdot )}(\mathbb R^{n})$, when $p(\cdot )\in \mathscr {P}(\mathbb R^{n})$, $0<{\beta }<n/p_{+}$ and $1/q(\cdot )=1/p(\cdot )-\beta /n$ with $q(\cdot )(n-\beta )/n \in \mathscr {B}(\mathbb R^{n})$.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4360. Commuting Toeplitz operators on the pluriharmonic Bergman space
- Creator:
- Lee, Young Joo
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Toeplitz operators and pluriharmonic Bergman space
- Language:
- English
- Description:
- We prove that two Toeplitz operators acting on the pluriharmonic Bergman space with radial symbol and pluriharmonic symbol respectively commute only in an obvious case.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public