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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 $R$?

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    (not a duplicate) but a [related question](http://math.stackexchange.com/questions/91754/proving-idempotent-characteristics-on-matrices-algebra)2011-12-15

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Let $k$ be a field, let $T$ be the $k$-algebra of $n\times n$ upper triangular matrices, and let $J\subset T$ be the ideal of strictly upper triangular matrices. Let us assume that $n>1$ to avoid trivialities.

Then $T/J$ is isormorphic to $\underbrace{k\times\cdots\times k}_{n\text{ factors}}$. In particular, $T/J$ has a basis of central idempotents. On the other hand, $T$ is a connected algebra: it has exactly one non-zero central idempotent, namely its unit. It is clear now that not all central idempotents in $T/J$ can be lifted to $R$.

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    what does basis of central indempotent of $\frac{T}{J}$ looklike? and what is connected algebra?2011-12-16