ON JACOBSON PROPERTY OF $\Gamma_N$-RINGS

Abstract

Let $M$ be a $\Gamma$-ring in the sense of Nobusawa. The ring $M_2=\left (\begin{array}{ll} R~~& \Gamma\\ M& \Gamma\\ \end{array}\right )$ was defined by Kyuno. Let $\cal{P}$ be a class of prime rings such that for every prime ring $R$ and any $0\neq e^2=e\in R,~R\in \cal{P}$ if and only if $eRe\in \cal{P}$. In this paper, the $\cal{P}$-Jacobson $\Gamma$-rings which include the Jacobson property and Brown-McCoy property as special case are defined. Relationships between $\cal{P}$-Jacobson properties of $\Gamma$-ring $M$ and the corresponding properties of $\Gamma_{n,m}$-ring $M_{m,n}$, the right operator ring $R$ of $\Gamma$-ring $M,~M$-ring $\Gamma$ and the ring $M_2$ are established.