Some notes on $GQN$ rings


Abstract: A ring $R$ is called ageneralized quasinormal ring (abbreviated as $GQN$ ring) if $ea∈N(R)$ for each $e∈ E(R)$ and $a∈ N(R)$. The class of $GQN$ rings is a proper generalization of quasinormal rings and $NI$ rings. Many properties of quasinormal rings are extended to $GQN$ rings. For a$GQN$ ring $R$ and $a∈ R$, it is shown that:1) if $a$ is a regular element, then $a$ is a strongly regular element;2) if $a$ is an exchange element, then $a$ is clean;3) if $R$ is a semiperiodic ring with $J(R)\neq N(R)$, then $R$ is commutative;4) if $R$ is an $MVNR$, then $R$ is strongly regular.

Keywords: $GQN$ rings, (von Neumann) regular elements, $NI$ rings, quasinormal rings, generalized $GQN$ rings, semiperiodic rings, exchange rings

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