Intereting Posts

If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$
Why is $\pi$ irrational if it is represented as $c/d$?
Intersection of Affine Subsets of a Vector Space.
Where are the values of the sine function coming from?
How to solve this sum limit? $\lim_{n \to \infty } \left( \frac{1}{\sqrt{n^2+1}}+\cdots+\frac{1}{\sqrt{n^2+n}} \right)$
Existence of irreducible polynomials over finite field
$\sum_{n=1}^\infty a_n<\infty$ if and only if $\sum_{n=1}^\infty \frac{a_n}{1+a_n}<\infty$
Strictly convex Inequality in $l^p$
looking for a technique to solve an indefinite integral of one over the square root of a cubic polynomial
Which are the mathematical problems in non-standard analysis? (If any)
Is $\mathbb{N}$ impossible to pin down?
Linear map $f:V\rightarrow V$ injective $\Longleftrightarrow$ surjective
What are the last two digits of $77^{17}$?
Applications of additive version of Hilbert's theorem 90
Is there a third dimension of numbers?

I’m trying to prove that if $M$ is a simple module over $M_n(D)$ where $D$ is a division algebra, then $M\cong D^n$. I know that if $M$ is a simple module over $R$ then it is isomorphic to $Rv=\{rv|r\in R\}$ for all $v\in M$. I’m trying to find a $M_n(D)$ module isomorphism $f:M_n(D)v\rightarrow D^n$. I’ve been told to try the map that maps $Xv\mapsto Xe_1$, where $e_1$ is the column vector with 1st entry 1 and the rest 0. I can prove that this is a homomorphism but I’m stuck on the injectivity. Any help would be well appreciated!

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- Show that $\gcd(a,bc)=1$ if and only if $\gcd(a,b)=1$ and $\gcd(a,c)=1$
- A finite commutative ring with the property that every element can be written as product of two elements is unital
- Group of even order contains an element of order 2

The kernel of the map is itself a submodule of M. It must be trivial because M is simple.

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