GENERALIZING $\pi$-REGULAR RINGS Article Swipe
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· 2015
· Open Access
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· DOI: https://doi.org/10.11650/tjm.19.2015.6236
· OA: W190011893
We introduce the class of weakly nil clean rings, as rings $R$ in which for\nevery $a \\in R$ there existan idempotent $e$ and a nilpotent $q$ such that $a-e-q\n\\in eRa$. Every weakly nil clean ring is exchange. Weakly nil clean rings contain\n$\\pi$-regular rings as a proper subclass, and these two classes coincide in the case\nwhen the ring has central idempotents, or has bounded index of nilpotence, or is a\nPI-ring. Weakly nil clean rings also properly encompass nil clean rings of Diesl\n[13]. The center of a weakly nil clean ring is strongly $\\pi$-regular, and\nconsequently, every weakly nil clean ring is a corner of a clean ring. These results\nextend Azumaya [3], McCoy [25], and the second author [33] to a wider class of\nringsand provide partial answers to some open questions in [13] and [33]. Some other\nproperties are studied and several examples are given as well.
