Completely Centrally Essential Rings

TL;DR

The paper studies completely centrally essential rings, showing that semiprimary ones are Lie nilpotent with class bounded by the nilpotence index of the Jacobson radical $J(R)$ and that noetherian ones are strongly Lie nilpotent and hence $PI$. It proves that the property is preserved under forming the classical ring of fractions: for any completely centrally essential ring $R$, the localization $Q_{\text{cl}}(R)$ is again completely centrally essential. Additionally, it proves that if $R$ is a commutative domain and $G$ is any group, then the group ring $RG$ is commutative whenever it is completely centrally essential. The results combine invariance arguments, Ore conditions, and structural group-ring analysis (including the role of Hamiltonian groups and the quaternion group $Q_8$) to characterize when central essentiality forces commutativity in group rings and to describe the Lie-theoretic and PI implications for the ring structures involved.

Abstract

A ring $R$ is said to be centrally essential if for every its non-zero element $a$, there exist non-zero central elements $x$ and $y$ with $ax = y$. A ring $R$ is said to be completely centrally essential if all its factor rings are centrally essential rings. It is proved that completely centrally essential semiprimary rings are Lie nilpotent; noetherian completely centrally essential rings are strongly Lie nilpotent (in particular, every such a ring is a $PI$-ring). Every completely centrally essential ring has the classical ring of fractions which is a completely centrally essential ring. If $R$ is a commutative domain and $G$ is an arbitrary group, then any completely centrally essential group ring $RG$ is commutative.