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Let $(A,m,k)$ be a local ring, and $A$ is a finitely generated $k$-algebra, and the maximal ideal $m$ is nilpotent. Let $x_1,\ldots,x_n \in m$ and their canonical images in $m/m^2$ generate this $k$-vector space.

How to show that $x_1,\ldots,x_n$ generate $A$ as $k$-algebra?

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    When you put a hyphen between dollar-signs, so it's inside of $\TeX$, then it looks like a minus sign instead of a hyphen. Not only is a minus sign longer, but it is preceded and followed by spaces that don't happen with a hyphen. Observe: $k-$algebra versus $k$-algebra.2012-03-22

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