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Gut Microbiome Functional Reprogramming Reflects Divergent Social Strategies in a Wild Primate
Integrative Zoology, EarlyView.Wild golden snub‐nosed monkeys employ divergent social strategies, forming reproductive one‐male units and bachelor all‐male units. We reveal that these social structures, intertwined with seasonal diets, drive distinct functional reprogramming in their gut microbiomes. This adaptive microbial plasticity plays a crucial role in primate socio‐ecological
Guanwei Lan +11 more
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Southeast Asian Bulletin of Mathematics, 2001
For a set of primes \(\pi\), the concept of Engel conditions for finite groups is extended. For example an element \(g\) is a weakly right \(\pi\)-Engel element if for each \(\pi'\)-element \(x\) in the group \(G\) there is a positive integer \(n\) such that the \((n+1)\)-commutator \([x,g,\dots,g]\) is a \(\pi\)-element.
Fan, Yun, Hai, Jinke
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For a set of primes \(\pi\), the concept of Engel conditions for finite groups is extended. For example an element \(g\) is a weakly right \(\pi\)-Engel element if for each \(\pi'\)-element \(x\) in the group \(G\) there is a positive integer \(n\) such that the \((n+1)\)-commutator \([x,g,\dots,g]\) is a \(\pi\)-element.
Fan, Yun, Hai, Jinke
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An Engel condition with skew derivations
Monatshefte für Mathematik, 2008The authors extend [\textit{C. Lanski}, Proc. Am. Math. Soc. 118, No. 3, 731-734 (1993; Zbl 0821.16037)] from derivations to skew derivations. Let \(R\) be a prime ring and \(L\) a noncommutative Lie ideal of \(R\). For \(x,y\in R\) set \([x,y]_1=[x,y]=xy-yx\) and when \(n>1\) let \([x,y]_n=[[x,y]_{n-1},y]\).
Chou, Ming-Chu, Liu, Cheng-Kai
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Engel Condition and p-nilpotency of Finite Groups
Acta Mathematica Sinica, English Series, 2021zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Wang, Lei +2 more
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Skew derivations with annihilating Engel conditions
Publicationes Mathematicae Debrecen, 2006Let \(R\) be a noncommutative prime ring. Let \(\sigma\) be an automorphism of \(R\), \(\delta\) be a \(\sigma\)-derivation, and \(a\in R\). The authors prove that if \(a[\delta(x),x]_k=0\) for any \(x\in R\), where \(k\) is a fixed positive integer, then either \(a=0\) or \(\delta=0\), except when \(R=M_2(\text{GF}(2))\).
Chuang, C. L., Chou, M. C., Liu, C. K.
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RINGS SATISFYING GENERALIZED ENGEL CONDITIONS
Journal of Algebra and Its Applications, 2012Let R be an associative ring and let x, y ∈ R. Define the generalized commutators as follows: [x, 0y] = x and [x, ky] = [x, k-1y]y - y[x, k-1y](k = 1, 2, …). In this paper we study some generalized Engel rings, i.e. [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)y] = 0), [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)yn(x, y)] = 0 ...
Ramezan-Nassab, M., Kiani, D.
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Power closure and the Engel condition
Israel Journal of Mathematics, 1997A Lie \(p\)-algebra \(L\) is called \(n\)-power closed if in every section of \(L\), any sum of two \(p^{i+n}\)-th powers is a \(p^i\)-th power \((i>0)\). The authors prove that if \(L\) is residually nilpotent and \(n\)-power closed for some \(n\geq 0\) then \(L\) is \((3p^{n+2}+1)\)-Engel if \(p\geq 2\) and \((3\cdot 2^{n+3}+1)\)-Engel if \(p=2 ...
Riley, David M., Semple, James F.
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On the Centralizers of Derivations with Engel Conditions
Communications in Algebra, 2013Let R be a noncommutative prime ring and d, δ two nonzero derivations of R. If δ([d(x), x] n ) = 0 for all x ∈ R, then char R = 2, d 2 = 0, and δ = αd, where α is in the extended centroid of R. As an application, if char R ≠ 2, then the centralizer of the set {[d(x), x] n | x ∈ R} in R coincides with the center of R.
Cheng-Kai Liu, Wen-Kwei Shiue
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On Artinian rings satisfying the Engel condition
Ukrainian Mathematical Journal, 2006Summary: Let \(R\) be an Artinian ring, not necessarily with a unit, and let \(R^\circ\) be the group of all invertible elements of \(R\) with respect to the operation \(a\circ b=a+b+ab\). We prove that the group \(R^\circ\) is a nilpotent group if and only if it is an Engel group and the quotient ring of the ring \(R\) by its Jacobson radical is ...
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ON n-ENGEL PAIR SATISFYING CERTAIN CONDITIONS
Journal of Algebra and Its Applications, 2014Let G be a group and h, g ∈ G. The 2-tuple (h, g) is said to be an n-Engel pair, n ≥ 2, if h = [h,n g], g = [g,n h] and h ≠ 1. In this paper, we prove that if (h, g) is an n-Engel pair, hgh-2gh = ghg and ghg-2hg = hgh, then n = 2k where k = 4 or k ≥ 6. Furthermore, the subgroup generated by {h, g} is determined for k = 4, 6, 7 and 8.
Quek, S. G., Wong, K. B., Wong, P. C.
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