arXiv subjects
Publications and source records attributed to Blake W. Madill.
Given a t.u.p.-semigroup (two unique product semigroup) $X$, we show if $R$ is an $X$-graded ring then both its nilradical and Jacobson radical are homogeneous. This partially answers questions of Smoktunowicz and Jespers.
Given a finitely generated free monoid $X$ and a morphism $ϕ: X\to X$, we show that one can construct an algebra, which we call an iterative algebra, in a natural way. We show that many ring theoretic properties of iterative algebras can be easily characterized in terms of linear algebra and combinatorial data from the morphism and that, moreover, it is decidable whether or not an iterative algebra has these properties. Finally, we use our construction to answer several questions of Greenfeld, Leroy, Smoktunowicz, and Ziembowski by constructing a primitive graded nilpotent algebra with Gelfand-Kirillov dimension two that is finitely generated as a Lie algebra.
Let $R$ be a ring satisfying a polynomial identity and let $D$ be a derivation of $R$. We consider the Jacobson radical of the skew polynomial ring $R[x;D]$ with coefficients in $R$ and with respect to $D$, and show that $J(R[x;D])\cap R$ is a nil $D$-ideal. This extends a result of Ferrero, Kishimoto, and Motose, who proved this in the case when $R$ is commutative.
Let $R$ be a ring satisfying a polynomial identity and let $δ$ be a derivation of $R$. We show that if $N$ is the nil radical of $R$ then $δ(N)\subseteq N$ and the Jacobson radical of $R[x;δ]$ is equal to $N[x;δ]$. As a consequence, we have that if $R$ is locally nilpotent then $R[x;δ]$ is locally nilpotent. This affirmatively answers a question of Smoktunowicz and Ziembowski.