ANNIHILATOR CONDITIONS IN MATRIX AND SKEW POLYNOMIAL RINGS

作者：Alhevaz A; Moussavi A^*

来源：Journal of Algebra and Its Applications, 2012, 11(4): 1250079.

DOI：10.1142/S021949881250079X

• Skew Armendariz ring
• skew polynomial ring
• Jacobson radical
• zip ring
• rigid ring
• upper triangular matrix ring

Let R be a ring with an endomorphism alpha and alpha-derivation delta. 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(n)x(n), g(x) = b(0) + b(1)x + ... + b(m)x(m) in R[x; alpha, delta], f(x)g(x) = 0 implies a(0)b(j) = 0 for each 0 %26lt;= j %26lt;= m. In this paper, radicals of the skew polynomial ring R[x; alpha, delta], in terms of a skew Armendariz ring R, is determined. We prove that several properties transfer between R and R[x; alpha, delta], in case R is an alpha-compatible skew Armendariz ring. We also identify some %26quot;relatively maximal%26quot; 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.

• 出版日期2012-8