A CONDITION FOR SIMPLE RING IMPLYING FIELD II

Authors

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

DOI:

https://doi.org/10.5556/j.tkjm.25.1994.4439

Keywords:

Jacobson radical, local ring, semiprime ring, simple ring, field

Abstract

It is shown that if $R$ is a simple ring with identity 1 and with a nonzero idempotent $e$ and satisfies the condition $(P_2)_e$ :

$(P_2)_e$    If $e- (a_1b_1+a_2b_2)$ is a right (left)zero divisor in $R$, then so is $e- (b_1a_1+b_2a_2)$.

then $R$ is a field.Thus if $R$ is a simple ring then $eRe$ is a field for every nonzero idempotent $e$ in $R$ if it exists and $eRe$ satisfies $(P_2)_e$. We also discuss the above property for the simple ring case by eliminating the identity 1.

References

I. N. Herstein, Rings with involution, Univ. of Chicago press (Chicago, 1976).

N. Jacobson, Basic Algebra. /, W. H. Freeman (San Francisco, 1974).

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

C. T. Yen, "A condition for simple ring implying field," unpublished manuscript.

C. T . Yen, "On rings satisfying both of 1 - abc and 1 - cba being invertible or none," Tamkang J. Math., 24, No. 3(1993), 317-321.

C. T. Yen, "A note on simple rings satisfying a condition," unpublished manuscript.

C. T. Yen, "A condition for simple ring implying field III," submitted.

C. T. Yen, "Associative rings with some conditions," unpublished manuscript.

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Published

1994-06-01

How to Cite

YEN, C.-T. (1994). A CONDITION FOR SIMPLE RING IMPLYING FIELD II. Tamkang Journal of Mathematics, 25(2), 163-166. https://doi.org/10.5556/j.tkjm.25.1994.4439

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Papers