# ON RINGS SATISFYING BOTH OF 1-abc AND 1-cba BEING INVERTIBLE OR NONE

## Authors

• CHEN-TE YEN Department of Mathematics, Chung Yuan University, Chung Li, Taiwan, 320, Republic of China.

## Keywords:

Regular, semisimple, prime ring, Perlis-Jacobson radical, directly finite

## Abstract

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.

## References

K. R. Goodearl, "Ring theory", Marcel Dekker Inc, New York and Basel, 1976.

I. N. Herstein, "Noncommutative rings", Carus Math. Monographs, No. 15, Math. Assoc. of Amer., 1971.

N. Jacobson, "Basic algebra 1", San Francisco, W. H. Freeman, 1974.

I. Kaplansky, "Fields and rings", Univ. of Chicago Press, Chicago, 1972.

J. Lambek, "Lecture on rings and modules", Waltham, Mass.: Blaisdell, 1966.

C. T. Yen, "On the commutativity of primary rings", Math. Japonica, 25 (1980), 449-452.

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C. T. Yen, "Another condition on simple ring implying field", unpublished manuscript.

1993-09-01

## How to Cite

YEN, C.-T. (1993). ON RINGS SATISFYING BOTH OF 1-abc AND 1-cba BEING INVERTIBLE OR NONE. Tamkang Journal of Mathematics, 24(3), 317-321. https://doi.org/10.5556/j.tkjm.24.1993.4503

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