Let $D$ be a division ring, and $M_i$ the set of matrices $n\times n$ with coefficients in $D$ with a nonzero $i$th column and zeros elsewhere. Then $M_i$ is simple for all $i$. how can I prove this?
why these modules are simple?
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abstract-algebra
noncommutative-algebra
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1$M_i$ is not a simple module over $D$. But it is simple over the ring of $n\times n$ matrices over $D$. – 2012-04-19
1 Answers
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Yes, mt_ is right. To elaborate: As a module over $D$, $M_i$ is the direct sum of $n$ copies of $D$, thus it is far from being simple.
To show that it is simple over $D^{n \times n}$, let $m \in M_i$ be an arbitrary element, $m \neq 0$. Then, it is not too hard to show that $(D^{n \times n}) m = M_i$ (since $D$ is a division ring).