Step in proof of Structure Theorem for finitely generated modules over a PID
I'm looking for help with problem #6 on page 189 of Jacobson's "Basic Algebra I". In particular, we suppose $D$ is a PID and $M$ is a finitely generated module over $D$ with generators $x_1, ... x_n$. Furthermore, we specify these generators have minimal $n$ and that $l(x_1)$ is minimal among generating sets with $n$ members. ($l(x)$ is defined to be number of primes in the factorization of $a in D$ where $text{ann}(x) = (a)$. If $a=0$ then we define $l(a) = infty$.)
Define $N := sum_{j ge 2}{Dx_j}$. The problem claims "To show $text{ann}(x_1) supset text{ann}(y)$ for $y in N$ it suffices to prove $text{ann}(x_1) supset text{ann}(x_j), j ge 2$". Can someone clarify for me why this is sufficient?
abstract-algebra modules principal-ideal-domains
asked Jan 4 at 0:52
Jeremiah Goertz
313
add a comment |
I'm looking for help with problem #6 on page 189 of Jacobson's "Basic Algebra I". In particular, we suppose $D$ is a PID and $M$ is a finitely generated module over $D$ with generators $x_1, ... x_n$. Furthermore, we specify these generators have minimal $n$ and that $l(x_1)$ is minimal among generating sets with $n$ members. ($l(x)$ is defined to be number of primes in the factorization of $a in D$ where $text{ann}(x) = (a)$. If $a=0$ then we define $l(a) = infty$.)
Define $N := sum_{j ge 2}{Dx_j}$. The problem claims "To show $text{ann}(x_1) supset text{ann}(y)$ for $y in N$ it suffices to prove $text{ann}(x_1) supset text{ann}(x_j), j ge 2$". Can someone clarify for me why this is sufficient?
abstract-algebra modules principal-ideal-domains
asked Jan 4 at 0:52
Jeremiah Goertz
313
If $din D$ kills $yin N$, then from the decomposition of $M$ into $Dx_j$'s $d$ must kill each component of $y$ in $Dx_j$. So it suffices to show that any $d$ killing $x_j$ kills $x_1$, which is the claim.
– cjackal
Jan 4 at 2:31
The "d must kill each component of y" isn't obvious. As stated in question's title, this is part of the proof of the structure theorem, so we can't assume the decomposition that theorem provides
– Jeremiah Goertz
2 days ago
(I.e., we can't assume the sum is direct, so it isn't immediate that $d$ must kill each component of $y$ in order to kill $y$.)
– Jeremiah Goertz
2 days ago
add a comment |
I'm looking for help with problem #6 on page 189 of Jacobson's "Basic Algebra I". In particular, we suppose $D$ is a PID and $M$ is a finitely generated module over $D$ with generators $x_1, ... x_n$. Furthermore, we specify these generators have minimal $n$ and that $l(x_1)$ is minimal among generating sets with $n$ members. ($l(x)$ is defined to be number of primes in the factorization of $a in D$ where $text{ann}(x) = (a)$. If $a=0$ then we define $l(a) = infty$.)
Define $N := sum_{j ge 2}{Dx_j}$. The problem claims "To show $text{ann}(x_1) supset text{ann}(y)$ for $y in N$ it suffices to prove $text{ann}(x_1) supset text{ann}(x_j), j ge 2$". Can someone clarify for me why this is sufficient?
abstract-algebra modules principal-ideal-domains
asked Jan 4 at 0:52
Jeremiah Goertz
313
I'm looking for help with problem #6 on page 189 of Jacobson's "Basic Algebra I". In particular, we suppose $D$ is a PID and $M$ is a finitely generated module over $D$ with generators $x_1, ... x_n$. Furthermore, we specify these generators have minimal $n$ and that $l(x_1)$ is minimal among generating sets with $n$ members. ($l(x)$ is defined to be number of primes in the factorization of $a in D$ where $text{ann}(x) = (a)$. If $a=0$ then we define $l(a) = infty$.)
Define $N := sum_{j ge 2}{Dx_j}$. The problem claims "To show $text{ann}(x_1) supset text{ann}(y)$ for $y in N$ it suffices to prove $text{ann}(x_1) supset text{ann}(x_j), j ge 2$". Can someone clarify for me why this is sufficient?
abstract-algebra modules principal-ideal-domains
abstract-algebra modules principal-ideal-domains
asked Jan 4 at 0:52
Jeremiah Goertz
313
asked Jan 4 at 0:52
Jeremiah Goertz
313
edited 2 days ago
asked Jan 4 at 0:52
Jeremiah Goertz
313
asked Jan 4 at 0:52
Jeremiah Goertz
313
asked Jan 4 at 0:52
Jeremiah Goertz
313
313
If $din D$ kills $yin N$, then from the decomposition of $M$ into $Dx_j$'s $d$ must kill each component of $y$ in $Dx_j$. So it suffices to show that any $d$ killing $x_j$ kills $x_1$, which is the claim.
– cjackal
Jan 4 at 2:31
The "d must kill each component of y" isn't obvious. As stated in question's title, this is part of the proof of the structure theorem, so we can't assume the decomposition that theorem provides
– Jeremiah Goertz
2 days ago
(I.e., we can't assume the sum is direct, so it isn't immediate that $d$ must kill each component of $y$ in order to kill $y$.)
– Jeremiah Goertz
2 days ago
add a comment |
If $din D$ kills $yin N$, then from the decomposition of $M$ into $Dx_j$'s $d$ must kill each component of $y$ in $Dx_j$. So it suffices to show that any $d$ killing $x_j$ kills $x_1$, which is the claim.
– cjackal
Jan 4 at 2:31
The "d must kill each component of y" isn't obvious. As stated in question's title, this is part of the proof of the structure theorem, so we can't assume the decomposition that theorem provides
– Jeremiah Goertz
2 days ago
(I.e., we can't assume the sum is direct, so it isn't immediate that $d$ must kill each component of $y$ in order to kill $y$.)
– Jeremiah Goertz
2 days ago
If $din D$ kills $yin N$, then from the decomposition of $M$ into $Dx_j$'s $d$ must kill each component of $y$ in $Dx_j$. So it suffices to show that any $d$ killing $x_j$ kills $x_1$, which is the claim.
– cjackal
Jan 4 at 2:31
If $din D$ kills $yin N$, then from the decomposition of $M$ into $Dx_j$'s $d$ must kill each component of $y$ in $Dx_j$. So it suffices to show that any $d$ killing $x_j$ kills $x_1$, which is the claim.
– cjackal
Jan 4 at 2:31
The "d must kill each component of y" isn't obvious. As stated in question's title, this is part of the proof of the structure theorem, so we can't assume the decomposition that theorem provides
– Jeremiah Goertz
2 days ago
The "d must kill each component of y" isn't obvious. As stated in question's title, this is part of the proof of the structure theorem, so we can't assume the decomposition that theorem provides
– Jeremiah Goertz
2 days ago
(I.e., we can't assume the sum is direct, so it isn't immediate that $d$ must kill each component of $y$ in order to kill $y$.)
– Jeremiah Goertz
2 days ago
(I.e., we can't assume the sum is direct, so it isn't immediate that $d$ must kill each component of $y$ in order to kill $y$.)
– Jeremiah Goertz
2 days ago
add a comment |
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If $din D$ kills $yin N$, then from the decomposition of $M$ into $Dx_j$'s $d$ must kill each component of $y$ in $Dx_j$. So it suffices to show that any $d$ killing $x_j$ kills $x_1$, which is the claim.
– cjackal
Jan 4 at 2:31
The "d must kill each component of y" isn't obvious. As stated in question's title, this is part of the proof of the structure theorem, so we can't assume the decomposition that theorem provides
– Jeremiah Goertz
2 days ago
(I.e., we can't assume the sum is direct, so it isn't immediate that $d$ must kill each component of $y$ in order to kill $y$.)
– Jeremiah Goertz
2 days ago