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The total graph and regular graph of a commutative ring J. Pure Appl. Alg. 213 2224 – 2228 10.1016/j.jpaa.2009.03.013 . [2
Abstract
Abstract
A ring R is called almost-quasi-commutative if for each x, y ∈ R there exist nonzero relatively prime integers j = j(x, y) and k = k(x, y) and a non-negative integer n = n(x, y) such that jxy = k(yx) n . We establish some general properties of such rings, study commutativity of almost-quasi-commutative R, and consider several examples.
-dimensional endo-commutative algebras over an arbitrary field . arXiv : 2304.00491v1 [math.RA] 2 Apr 2023 . [3] Goze , M . and Remm , E . 2-dimensional algebras . Afr. J. Math. Phys . 10 , 1 ( 2011 ), 81 – 91 . [4] Kaygorodov , I . and Volkov
. 16 #R117. [2] Anderson , D. F. Badawi , A. 2008 The total graph of a commutative ring J. Algebra
Abstract
I. N. Herstein [10] proved that a prime ring of characteristic not two with a nonzero derivation d satisfying d(x)d(y) = d(y)d(x) for all x, y must be commutative, and H. E. Bell and M. N. Daif [8] showed that a prime ring of arbitrary characteristic with nonzero derivation d satisfying d(xy) = d(yx) for all x, y in some nonzero ideal must also be commutative. For semiprime rings, we show that an inner derivation satisfying the condition of Bell and Daif on a nonzero ideal must be zero on that ideal, and for rings with identity, we generalize all three results to conditions on derivations of powers and powers of derivations. For example, let R be a prime ring with identity and nonzero derivation d, and let m and n be positive integers such that, when charR is finite, m ∨ n < charR. If d(x m y n ) = d(y n x m ) for all x, y ∈ R, then R is commutative. If, in addition, charR≠ 2 and the identity is in the image of an ideal I under d, then d(x) m d(y) n = d(y) n d(x) m for all x, y ∈ I also implies that R is commutative.
We give a characterization of all those commutative groups which admit at least one absolutely nonmeasurable homomorphism into the real line (or into the one-dimensional torus). These are exactly those commutative groups (G, +) for which the quotient group G/G 0 is uncountable, where G 0 denotes the torsion subgroup of G.
Abstract
For a finite abelian group A, the ring End(A) is a maximal ring in the nearring M(A). In this paper we consider an analogous problem for finite semigroups, concentrating on commutative Clifford semigroups.
