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1.
Let A be a prime ring with nonzero right ideal R and f : R → A an additive map. Next, let k,n1, n2,…,nk be natural numbers. Suppose that […[[(x), xn1], xn2],…, xnk]=0 for all x ∈ R. Then it is proved in Theorem 1.1 that [f(x),x]=0 provided that either char(A)=0 or char (A)> n1+n2+ …+nk Theorem 1.1 is a simultaneous generalization of a number of results proved earlier. 相似文献
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Shuliang Huang 《Czechoslovak Mathematical Journal》2011,61(4):1135-1140
Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d[x, y])
m
= [x, y]
n
for all x, y ∈ I, then R is commutative. (ii) If Char R ≠ 2 and [d(x), d(y)]
m
= [x, y]
n
for all x, y ∈ I, then R is commutative. Moreover, we also examine the case when R is a semiprime ring. 相似文献
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In 1992, Wilson and Zelmanov proved that a profinite Engel group is locally nilpotent. Here we prove the stronger result that
every compact Engel group is locally nilpotent. 相似文献
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M. Mahdavi-Hezavehi S. Akbari-Feyzaabaadi M. Mehraabaadi H. Hajie-Abolhassan 《代数通讯》2013,41(8):2881-2887
Let D be a division ring with centre F and denote by D′ the derived group (commutator subgroup) of D ? = D ? {0}. It is shown that if each element of D′ is algebraic over F, then D is algebraic over F. It is also proved that each finite separable extension of F in D is of the form F(c) for some element c in the derived group D′. Using these results, it is shown that if each element of the derived group D′ is of bounded degree over F, then D is finite dimensional over F. 相似文献
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Pavel Shumyatsky 《代数通讯》2013,41(4):1937-1940
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Roger Yue Chi Ming 《Annali di Matematica Pura ed Applicata》1984,138(1):245-253
Summary
In this sequel to [14]and [15],a generalization of quasi-injective modules, noted FK-injective, is introduced to study von Neumann regular and continuous rings. This will lead to new characteristic properties of continuous regular rings. Conditions for certain non-singular modules to be completely reducible and injective are given. A few decompositions of FK-injective modules are considered.Dedicated to Professor Hisao Tominaga 相似文献
15.
If (R,M) is a quasilocal integral domain of Krull dimension n,?1<n≤∞, and E is the direct sum of denumerably many copies of R/M, then T:=R?E is a reduced n-dimensional universal survival ring which is not a universal lying-over ring. In fact, T is a new kind of such ring, as T is not (isomorphic to) an A+B construction and T is not a ring of continuous real-valued functions. The analysis includes identifying all the prime ideals of T and showing that T is its own total quotient ring and satisfies Property A. The assertion would fail if n=1, as T would be a universal lying-over ring in this case. It is also shown that a (commutative unital) ring A satisfies Property A if and only if each ideal of A that consists only of zero-divisors survives in the complete ring of quotients of A. 相似文献
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Abstract In this paper we introduce generalized ideal-stable regular rings. It is shown that if a regular ring R is a generalized I-stable ring, then every square matrix over I is the product of an idempotent matrix and an generalized invertible matrix and admits a diagonal reduction by some generalized invertible matrices. 相似文献
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A principal right ideal of a ring is called uniquely generated if any two elements of the ring that generate the same principal right ideal must be right associated (i.e., if for all a,b in a ring R, aR = bR implies a = bu for some unit u of R). In the present paper, we study “uniquely generated modules” as a module theoretic version of “uniquely generated ideals,” and we obtain a characterization of a unit-regular endomorphism ring of a module in terms of certain uniquely generated submodules of the module among some other results: End(M) is unit-regular if and only if End(M) is regular and all M-cyclic submodules of a right R-module M are uniquely generated. We also consider the questions of when an arbitrary element of a ring is associated to an element with a certain property. For example, we consider this question for the ring R[x;σ]∕(xn+1), where R is a strongly regular ring with an endomorphism σ be an endomorphism of R. 相似文献