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Start Over Coverage 791-801

1. A class of statistical and $\sigma$-conservative matrices

Creator:
Çoşkun, Hüsamettin and Çakan, Celal
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
statistical convergence, invariant means, core theorems, and matrix transformations
Language:
English
Description:
In [5] and [10], statistical-conservative and $\sigma $-conservative matrices were characterized. In this note we have determined a class of statistical and $\sigma $-conservative matrices studying some inequalities which are analogous to Knopp’s Core Theorem.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. 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