Let $R$ be a ring, let $I$ be an ideal of $R$, and let $u\in R$ be idempotent modulo $I$ (that is, $u^{2}-u\in I$). Then $u$ can be lifted to an idempotent in $R$ in case there is an idempotent $e$ in $R$ with $e-u \in I$.
I want to know that if $R$ is the ring of $n\times n$ upper triangular matrice over a field $\mathbb{Q}$ and if $J$ is the ideal of matrices having zero on the diagonal, then there are idempotents that are central modulo (in the centre) $J$ that can not be lifted to central idempotents of $R$?