| 1. | The endomorphism ring is simply the ring of formal power series.
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| 2. | Any such complex torus has the Gaussian integers as endomorphism ring.
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| 3. | This is an example of an endomorphism that is not an automorphism.
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| 4. | The characteristic polynomial of this endomorphism has the following form:
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| 5. | Consequently, the endomorphism ring of any simple module is a division ring.
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| 6. | Square matrix rings arise as endomorphism rings of free modules with finite rank.
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| 7. | In particular, if is a field then the Frobenius endomorphism is injective.
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| 8. | An automorphism is a morphism that is both an endomorphism and an isomorphism.
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| 9. | In particular, the endomorphism ring of a simple module is a division ring.
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| 10. | Otherwise ? is an endomorphism but not a ring " automorphism ".
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