A CONDITION FOR SIMPLE RING IMPLYING FIELD II
Keywords:Jacobson radical, local ring, semiprime ring, simple ring, field
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.
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