Bulletin of the
Korean Mathematical Society BKMS

This paper is motivated by the results in [2], [10], [13] and [19]. We study some properties of generalizations of commutative rings and relations between them. We also show that for a right quasi-duo right weakly $\pi$-regular ring $R$, $R$ is an $(S,2)$-ring if and only if every idempotent in $R$ is a sum of two units in $R$, which gives a generalization of [2, Theorem 4] on right quasi-duo rings. Moreover we find a condition which is equivalent to the strongly $\pi$-regularity of an abelian right quasi-duo ring.

Keywords: quasi-duo ring, $\pi$-regular ring, $(S,2)$-ring and 2-primal ring

MSC numbers: 16D25, 16D30, 16N20, 16E50