Annihilator conditions in matrix and skew polynomial rings. (Q2909810)

Let \(R\) be a ring with 1, an endomorphism \(\alpha\) of \(R\), and an \(\alpha\)-derivation \(\delta\). The Ore extension of \(R\), \(R[x;\alpha,\delta]\) is called a skew Armendariz ring if for \(f(x)=\sum_{i=0}^na_ix^i\), \(g(x)=\sum_{j=0}^mb_jx^j\), \(f(x)g(x)=0\) implies \(a_0b_j=0\) for each \(j\). For \(n=m=1\), if \(f(x)g(x)=0\) implies \(a_0b_1=a_1b_0=0\), then \(R\) is called linearly Armendariz. An \(\alpha\) is called compatible if for all \(a,b\in R\), \(ab=0\) if and only if \(a\alpha(b)=0\); and \(R\) is called \(\alpha\) compatible if \(R\) has a compatible \(\alpha\). If \(ab=0\) implies \(a\delta(b)=0\), then \(R\) is called \(\delta\) compatible; and \(R\) is \((\alpha,\delta)\) compatible if it is both \(\alpha\) and \(\delta\) compatible.NEWLINENEWLINE Let \(R\) be an \(\alpha\) compatible linearly skew Armendariz ring. The authors show that the following radicals of \(R\) are the same: \(N_0(R)=\text{Nil}_*(R)=\text{L-rad}(R)=\text{Nil}^*(R)=A(R)\) where \(N_0(R)\) is the Wedderburn radical of \(R\), \(\text{Nil}_*(R)\) the lower nil radical, \(\text{L-rad}(R)\) the Levitzki radical, \(\text{Nil}^*(R)\) the upper nil radical, and \(A(R)\) the sum of all nil left ideals of \(R\). Also \(J(R[x;\alpha,\delta])\cap R\) is a nil ideal of \(R\) where \(J(R[x;\alpha,\delta])\) is the Jacobson radical of \(R[x;\alpha,\delta]\). Moreover, the above equations of radicals also hold for \(R[x;\alpha,\delta]\) over an \(\alpha\) compatible skew Armendariz ring \(R\).NEWLINENEWLINE Let \(T_n(R)\) be the upper triangular matrix ring of order \(n\) for some integer \(n\) over \(R\) with the extended endomorphism \(\alpha\) from \(R\) and the extended derivation \(\delta\). Then there exists a linearly skew Armendariz ring \(R\) such that \(T_n(R)\) is not linearly skew Armendariz. The authors show some maximal skew Armendariz subrings of \(T_n(R)\) over an \(\alpha\) rigid and reduced ring \(R\) where \(\alpha\) is a monomorphism and \(\delta\) is an \(\alpha\)-derivation of \(R\).