Abstract
In this paper, we prove that if R is a Min-E ring, then the following statements are equivalent: (1) Every left primitive factor ring of R is left artinian; (2) R is a π-regular ring; (3) R is an Exchange ring; (4) R is a Clean ring. As an application, we obtain that if R is Min-E ring, every left primitive factor ring of R is artinian and any direct product of them is a Min-E ring, then every non-zero homomorphic image of R contains only a finite number of prime ideals. When R is Min-E ring and has no infinite set of orthogonal idempotents, a left R-module M is Min-E if and only if M is Max-E. Also, we show, if R is a strongly clean Min-E ring, then R is directly finite and has stable range 1.
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AMS Subject Classification (1991) 16A30 16P20
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Chen, MS., Li, JQ. On Rings Whose Primitive Factor Rings Are Left Artinian. SEA bull. math. 25, 23–30 (2001). https://doi.org/10.1007/s10012-001-0023-3
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DOI: https://doi.org/10.1007/s10012-001-0023-3