How to express, the two sided principal ideal of a ring without unity(Rng)?
How to express, the two sided ideal of a ring $R$ generated by some subset $S\subseteq R$ where $R$ is ring without unity(Rng).
My attempt: For first one, when $R$ is ring with unity then the two sided principal ideal of $R$ is given by
$RaR=\{r_1as_1+...+r_nas_n : r_1,s_1,...r_n,s_n\in R\},$
For some element $a\in R$.
But, when $R$ is ring without unity then $a$ may not belongs to $RaR$. So how to write/represent the two sided principal ideal in this case? Is it is given by $RaR+a\mathbb{Z}$ (am i correct?).
Further what about second question? Please help....