On differential polynomial rings over nil algebras
One of the many equivalent statements for the challenging Koethe conjecture is whether a polynomial ring \(R[X]\) is Jacobson radical if the ring \(R\) is nil. \textit{A. Smoktunowicz} [J. Algebra 233, No. 2, 427--436 (2000; Zbl 0969.16006)] has shown that \(R[x]\) need not be nil if the ring \(R\) is nil. Earlier \textit{E. R. Puczyłowski} and \textit{A. Smoktunowicz} [Commun. Algebra 26, No. 8, 2473--2482 (1998; Zbl 0909.16015)] showed that if \(R\) is nil, then \(R[X]\) is Brown-McCoy radical, i.e., \(R[X]\) cannot be mapped homomorphically onto a unital simple ring. It is natural to ask about the validity of similar statements for differential polynomial rings over nil rings. Let \(\delta :R\rightarrow R\) be a derivation of a ring \(R\). Then \(R[X;\delta]\) is the differential polynomial ring, that is the set of left polynomials \(a_{0}+a_{1}X+\dots+a_{n-1}X_{n-1}+a_{n}X_{n}\) with \(a_{i}\in R\) for all \(i\). Addition of such polynomials is defined in the usual way and multiplication is defined by the associative rule, converting right polynomials into left polynomials by repeatedly applying the rule \(Xa=aX+a^{\delta}\) for all \(a\in R\). Here, \(a^{\delta}\) denotes the image of \(a\) by \(\delta\). It is known that if \(R\) is locally nilpotent, then \(R[X;\delta]\) need not be Jacobson radical but it must be Brown-McCoy radical, hence it cannot be mapped homomorphically onto a simple ring with unity. Here the authors show that if \(R\) is a nil algebra over a field of characteristic \(0\) , then the differential polynomial ring \(R[X,\delta]\) has no homomorphic image which is a unital simple ring.
scientific article; zbMATH DE number 7318484
scientific article