Finding the injective hull

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Let's suppose that I have an element $e$ of order $p$ in the group of complex numbers whose elements all have order $p^n$ for some $ninmathbbN$ (henceforth called $K$), and the module generated by $(e)$ is irreducible.

How do I show that the injective hull of the module generated by $(e)$ is in fact, equal to $K$?

abstract-algebra ring-theory modules

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edited Jul 27 at 22:40

asked Jul 27 at 22:20

Tomislav Ostojich

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Let's suppose that I have an element $e$ of order $p$ in the group of complex numbers whose elements all have order $p^n$ for some $ninmathbbN$ (henceforth called $K$), and the module generated by $(e)$ is irreducible.

How do I show that the injective hull of the module generated by $(e)$ is in fact, equal to $K$?

abstract-algebra ring-theory modules

share|cite|improve this question

edited Jul 27 at 22:40

asked Jul 27 at 22:20

Tomislav Ostojich

475313

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

Let's suppose that I have an element $e$ of order $p$ in the group of complex numbers whose elements all have order $p^n$ for some $ninmathbbN$ (henceforth called $K$), and the module generated by $(e)$ is irreducible.

How do I show that the injective hull of the module generated by $(e)$ is in fact, equal to $K$?

abstract-algebra ring-theory modules

share|cite|improve this question

edited Jul 27 at 22:40

asked Jul 27 at 22:20

Tomislav Ostojich

475313

Let's suppose that I have an element $e$ of order $p$ in the group of complex numbers whose elements all have order $p^n$ for some $ninmathbbN$ (henceforth called $K$), and the module generated by $(e)$ is irreducible.

How do I show that the injective hull of the module generated by $(e)$ is in fact, equal to $K$?

abstract-algebra ring-theory modules

share|cite|improve this question

edited Jul 27 at 22:40

asked Jul 27 at 22:20

Tomislav Ostojich

475313

share|cite|improve this question

share|cite|improve this question

edited Jul 27 at 22:40

edited Jul 27 at 22:40

edited Jul 27 at 22:40

asked Jul 27 at 22:20

Tomislav Ostojich

475313

asked Jul 27 at 22:20

Tomislav Ostojich

475313

asked Jul 27 at 22:20

Tomislav Ostojich

475313

475313

add a comment | 

add a comment | 

1 Answer
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Hint: The abelian group of complex numbers that are $p^n$-th roots of unity for some $n$ is isomorphic to $mathbb Zleft[frac1p^nright]/mathbb Z$, with the submodule generated by the $p$-th roots of unity corresponding to $mathbb Zfrac1p$. Now the strategy is exactly the same as for Injective hull of the trivial k[x]-module (can you see the common generalization of these examples?).

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answered Jul 28 at 10:25

Hanno

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote



accepted

Hint: The abelian group of complex numbers that are $p^n$-th roots of unity for some $n$ is isomorphic to $mathbb Zleft[frac1p^nright]/mathbb Z$, with the submodule generated by the $p$-th roots of unity corresponding to $mathbb Zfrac1p$. Now the strategy is exactly the same as for Injective hull of the trivial k[x]-module (can you see the common generalization of these examples?).

share|cite|improve this answer

answered Jul 28 at 10:25

Hanno

14.4k21541

add a comment | 

up vote
1
down vote



accepted

Hint: The abelian group of complex numbers that are $p^n$-th roots of unity for some $n$ is isomorphic to $mathbb Zleft[frac1p^nright]/mathbb Z$, with the submodule generated by the $p$-th roots of unity corresponding to $mathbb Zfrac1p$. Now the strategy is exactly the same as for Injective hull of the trivial k[x]-module (can you see the common generalization of these examples?).

share|cite|improve this answer

answered Jul 28 at 10:25

Hanno

14.4k21541

add a comment | 

up vote
1
down vote



accepted

up vote
1
down vote



accepted

Hint: The abelian group of complex numbers that are $p^n$-th roots of unity for some $n$ is isomorphic to $mathbb Zleft[frac1p^nright]/mathbb Z$, with the submodule generated by the $p$-th roots of unity corresponding to $mathbb Zfrac1p$. Now the strategy is exactly the same as for Injective hull of the trivial k[x]-module (can you see the common generalization of these examples?).

share|cite|improve this answer

answered Jul 28 at 10:25

Hanno

14.4k21541

Hint: The abelian group of complex numbers that are $p^n$-th roots of unity for some $n$ is isomorphic to $mathbb Zleft[frac1p^nright]/mathbb Z$, with the submodule generated by the $p$-th roots of unity corresponding to $mathbb Zfrac1p$. Now the strategy is exactly the same as for Injective hull of the trivial k[x]-module (can you see the common generalization of these examples?).

share|cite|improve this answer

answered Jul 28 at 10:25

Hanno

14.4k21541

share|cite|improve this answer

share|cite|improve this answer

answered Jul 28 at 10:25

Hanno

14.4k21541

answered Jul 28 at 10:25

Hanno

14.4k21541

answered Jul 28 at 10:25

Hanno

14.4k21541

14.4k21541

add a comment | 

add a comment | 

 

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