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.
WANG, LONG and WEI, JUNCHAO
"Some notes on $GQN$ rings,"
Turkish Journal of Mathematics: Vol. 41:
6, Article 12.
Available at: https://journals.tubitak.gov.tr/math/vol41/iss6/12