# Right ideals of a ring $R$ with J(R) nil

Assume that $R$ is a unital ring with nil Jacobson radical $J(R)$. If each right ideal of $R/J(R)$ is idempotent, is it necessarily true that each right ideal of $R$ is idempotent or nil?

If $I$ is a right ideal of $R$ contained in $J(R)$ then the result follows. If $I$ is a right ideal of $R$ containing $J(R)$ and $x\in I$ then $x+ J(R)=∑x_t y_t +J(R)$ (a finite sum over $t$ with $x_t,y_t\in I$). Hence $x-∑x_t y_t$ is a nilpotent element as it belongs to $J(R)$...

That is all my try! Thanks for any pushing suggestion!

Let $F$ be a field, $S=F\times F$, and $M=F\times\{0\}$ and use the trivial extension $R=S\times M$ where $(s,m)+(s',m')=(s+s',m+m')$ and $(s,m)(s',m')=(ss', sm'+ms')$.
Clearly $J(R)=\{0\}\times M$ is nil (in fact it squares to zero.) Also $R/J(R)\cong F\times F$ has all right ideals idempotent.
Now look at $I:=(\{0\}\times F, M)\lhd R$. It's not nil (since it's not contained in $J(R)$) and it's not idempotent (since $I^2=(\{0\}\times F,\{0\})$.)
• I guess any right ideal of your $R$ is the sum of a nil and an idempotent. Right? – karparvar Aug 4 '16 at 22:08