On left \(T\)-nilpotent rings (Q6542427)

A ring \(R\) is a sum of two subrings, \(R=R_1+R_2\), if every element of \(R\) is a sum of an element of \(R_1\) and an element of \(R_2\). A main theme in this subject, is when do properties on \(R_1\) and \(R_2\) carry over to~\(R\)? The introduction of this paper gives a summary of results in this area.\N\NA ring is \textit{left \(T\)-nilpotent\/} if given any infinite sequence of elements \(\{a_n\}\) some product \(a_1\cdots a_n\) must be~0. The main result of this paper is that if \(R_1\) are \(R_2\) are left \(T\)-nilpotent, so is \(R=R_1+R_2\).\N\NThe authors next consider a generalization to rings \(R\) graded by a semigroup~\(S\), \(R=\oplus_{s\in S} R_s\), each \(R_sR_t\subseteq R_{st}\). Say that a sequence \(\{s_n\}\) of elements of~\(S\) has condition \((q)\) if whenever \(s=s_k\cdots s_\ell=s_{\ell+1}\cdots s_m\), \(s\) is not idempotent, \(s\ne s^2\). Then the property that each graded component \(R_s\) is left \(T\)-nilpotent necessarily implies that \(R\) is, occurs when \(S\) has no sequence satisfying condition~\((q)\).\N\NFinally, they generalize from \(S\)-gradings to \(S\)-sums, a weaker condition than that of being \(S\)-graded. \(R=\sum_{s\in S} R_s\) may not be a direct sum, and instead of \(R_sR_t\) being contained in \(R_{st}\), it is only required to be in \(R_u\), where \(u\) lies in the semigroup generated by \(st\). Then the left \(T\)-nilpotency of each \(R_s\) implies that of \(R\) precisely when \(S\) contains no 2-groups and has no sequence satisfying condition~\((q)\).