On rings whose essential maximal right ideals are GP-injective
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In this paper, we continue to study the von Neumann regularity of rings whose essential maximal right ideals are GP-injective. It is proved that the following statements are equivalent: (1) $R$ is strongly regular; (2) $R$ is a 2-primal ring whose essential maximal right ideals are GP-injective; (3) $R$ is a right (or left) quasi-duo ring whose essential maximal right ideals are GP-injective. Moreover, it is shown that $R$ is strongly regular if and only if $R$ is a strongly right (or left) bounded ring whose essential maximal right ideals are GP-injective. Finally, we prove that a PI-ring whose essential maximal right ideals are GP-injective is strongly $\pi$-regular.
Abstract
In this paper, we continue to study the von Neumann regularity of rings whose essential maximal right ideals are GP-injective. It is proved that the following statements are equivalent: (1) $R$ is strongly regular; (2) $R$ is a 2-primal ring whose essential maximal right ideals are GP-injective; (3) $R$ is a right (or left) quasi-duo ring whose essential maximal right ideals are GP-injective. Moreover, it is shown that $R$ is strongly regular if and only if $R$ is a strongly right (or left) bounded ring whose essential maximal right ideals are GP-injective. Finally, we prove that a PI-ring whose essential maximal right ideals are GP-injective is strongly $\pi$-regular.
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