Jacobson radicals of ring extensions.
In this paper \(R\) is an algebra with multiplicative identity over a field \(K\). If \(\sigma\) is a \(K\)-linear automorphism of \(R\), then a \(\sigma\)-derivation \(d\) is a \(K\)-linear map \(d\colon R\to R\) such that \(d(rs)=d(r)s+\sigma(r)d(s)\) for all \(r,s\in R\). The ring \(R^d=\{r\in R:d(r)=0\}\) is called the ring of constants. A \(\sigma\)-derivation \(d\) is said to be locally nilpotent if for every \(r\in R\), there exists \(n=n(r)\geq 1\) such that \(d^n(r)=0\). If \(q\) is a nonzero element of \(K\), the authors call their \(\sigma\)-derivation \(q\)-skew if \(d\sigma(r)=q\sigma d(r)\) for all \(r\in R\). For subsets \(A,B\) of a ring \(R\), let \(\text{r.ann}_A(B)=\{a\in A:Ba=0\}\). \(R\) is said to satisfy the acc on right annihilators of powers if, for every \(r\in R\), there exists \(n\geq 1\) such that \(\text{r.ann}_R(r)\subseteq\text{r.ann}_R(r^2)\subseteq\cdots\subseteq\text{r.ann}_R(r^n)=\text{r.ann}_R(r^{n+1})=\cdots\). If \(0\neq r\in R\), \(r\) is called normalizing if \(rR=Rr\). An automorphism \(\sigma\) is said to have locally finite order if, for every \(r\in R\), there exists \(n\geq 1\) such that \(\sigma^n(r)=r\). The construction of the skew polynomial ring \(R[x;\delta]\) of automorphism type using a single derivation can be extended to construct the smash product \(R\#U(L)\), where \(L\) is a Lie algebra acting on \(R\) as derivations and \(U(L)\) is the universal enveloping algebra of \(L\). \(J(R[x;\delta])\) denotes the Jacobson radical of \(R[x;\delta]\). In the first part of this paper the authors extend existing results on the Jacobson radical of skew polynomial rings of derivation type when the base ring has no nonzero nil ideals. In particular, they show that if \(R\) is an algebra with no nonzero nil ideals satisfying the acc on right annihilators of powers, then \(J(R[x;\delta])=0\) if \(\delta\) is a derivation on \(R\). Moreover, \(J(R\#U(L))=0\) if \(L\) is a Lie algebra acting on \(R\) as derivations. They also show that if \(R\) is a semiprime algebra whose every nonzero ideal contains a normalizing element, then \(J(R[x;\delta])=0\) if \(\delta\) is a derivation on \(R\). Moreover, \(J(R\#U(L))=0\) if \(L\) is a Lie algebra acting on \(R\) as derivations. In the second part of the paper, the authors examine algebras \(R\) with a locally nilpotent \(q\)-skew \(\sigma\)-derivation \(d\) focussing on conditions on \(R^d\) that guarantee that the Jacobson radical \(J(R)\) of \(R\) is zero. In particular, the authors show that if \(R\) is an algebra with a locally nilpotent regular \(q\)-skew \(\sigma\)-derivation \(d\), where either \(q\) is not a root of \(1\) or \(R\) has characteristic \(0\) and \(q=1\), then \(J(R)=0\) if \(R^d\) is semiprime Goldie. They also show that if \(R\) is an algebra with a locally nilpotent \(q\)-skew \(\sigma\)-derivation \(d\), where either \(q\) is not a root of \(1\) or \(R\) has characteristic \(0\) and \(q=1\), then \(J(R)=0\) if \(R^d\) is Goldie with \(J(R^d)=0\). In the third part of the paper, the authors restrict their work to algebras of characteristic \(0\). They show that if \(R\) is an algebra of characteristic \(0\) with a locally nilpotent regular \(\sigma\)-derivation \(d\) such that \(d\sigma=\sigma d\) and if \(R^d\) has no nonzero nil ideals, then \(J(R)=0\) in all of the following cases: (1) \(d\) is a derivation and \(R^d\) satisfies the acc on right annihilators of powers; (2) \(\sigma\) has locally finite order, \(R\) is an algebra over an uncountable field, and \(R^d\) satisfies the acc on right annihilators of powers; (3) \(\sigma\) has locally finite order and \(R^d\) satisfies the acc on right annihilators; (4) \(\sigma\) has locally finite order and \(R^d\) is reduced; (5) \(\sigma\) has locally finite order and \(R^d\) satisfies a polynomial identity. -- The authors conclude the paper with showing that if \(R\) is an algebra of characteristic \(0\) with a locally nilpotent \(\sigma\)-derivation \(d\) such that \(d\sigma=\sigma d\) and if \(J(R^d)=0\), then \(J(R)=0\) in all of the following cases: (1) \(d\) is a derivation and \(R^d\) satisfies the acc on right annihilators of powers; (2) \(\sigma\) has locally finite order, \(R\) is an algebra over an uncountable field, and \(R^d\) satisfies the acc on right annihilators of powers; (3) \(\sigma\) has locally finite order and \(R^d\) satisfies the acc on right annihilators; (4) \(\sigma\) has locally finite order and \(R^d\) is reduced; (5) \(\sigma\) has locally finite order and \(R^d\) satisfies a polynomial identity.
skew polynomial rings of derivation type
scientific article; zbMATH DE number 6141847