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首页> 外文期刊>Journal of algebra and its applications >Annihilator conditions in matrix and skew polynomial rings
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Annihilator conditions in matrix and skew polynomial rings

机译：矩阵和偏多项式环中的灭条件

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Let R be a ring with an endomorphism α and α-derivation δ. By [A. R. Nasr-Isfahani and A. Moussavi, Ore extensions of skew Armendariz rings, Comm. Algebra 36(2) (2008) 508-522], a ring R is called a skew Armendariz ring, if for polynomials f(x) = a _0 + a _1 x + □ + a _nx ~n, g(x) = b _0+b _1x + □ + b _mx ~m in R[x; α, δ], f(x)g(x) = 0 implies a _0b _j = 0 for each 0 ≤ j ≤ m. In this paper, radicals of the skew polynomial ring R[x; α, δ], in terms of a skew Armendariz ring R, is determined. We prove that several properties transfer between R and R[x; α, δ], in case R is an α-compatible skew Armendariz ring. We also identify some "relatively maximal" skew Armendariz subrings of matrix rings, and obtain a necessary and sufficient condition for a trivial extension to be skew Armendariz. Consequently, new families of non-reduced skew Armendariz rings are presented and several known results related to Armendariz rings and skew polynomial rings will be extended and unified.
机译：令R为具有内同态α和α-衍生δ的环。由[A. R. Nasr-Isfahani和A.Moussavi，歪斜的Armendariz环的矿石延伸，Comm。代数36（2）（2008）508-522]中，如果多项式f（x）= a _0 + _1 x +□+ a _nx〜n，g（x）= R [x中的b _0 + b _1x +□+ b _mx〜m α，δ]，f（x）g（x）= 0意味着对于每个0≤j≤m，_0b _j = 0。本文研究了偏多项式环R [x;根据偏斜的Armendariz环R确定α，δ]。我们证明了R和R [x;之间的几个特性转移。如果R是一个与α相容的斜Armendariz环，则α，δ]。我们还确定矩阵环的一些“相对最大”偏斜的Armendariz子环，并获得使平凡扩展成为偏斜的Armendariz的必要和充分条件。因此，提出了新的非减少偏斜Armendariz环族，并将扩展和统一与Armendariz环和偏多项式环有关的几个已知结果。

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