ON RINGS SATISFYING BOTH OF 1-abc AND 1-cba BEING INVERTIBLE OR NONE
DOI:
https://doi.org/10.5556/j.tkjm.24.1993.4503Keywords:
Regular, semisimple, prime ring, Perlis-Jacobson radical, directly finiteAbstract
Let $R$ be a ring with identity 1 and $n$ a positive integer. We define the property Pn as follows:
(Pn) If $1- a_1a_2a_3 \cdots a_{n-1}a_n$ is invertible in $R$, then so is $1- a_na_2a_3 \cdots a_{n-1}a_1$.
Thus, $R$ satisfies (Pn), for some $n \ge 3$ if and only if $R$ satisfies (P3). Some properties of rings satisfying (P3) are obtained, e.g., $R$ must be directly finite.
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C. T. Yen, "A condition for simple ring implying field", Acta Mathematica Hungarica, 61(1993), 51-52.
C. T. Yen, "Another condition on simple ring implying field", unpublished manuscript.
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