$mathbb {Z}_{84}/(7) cong mathbb {Z}_{7}$
$begingroup$
Prove $mathbb {Z}_{84}/(7) cong mathbb {Z}_{7}$ using each of the three isomorphism theorems for rings.
For the first isomorphism theorem I defined a homomorphism $phi: mathbb {Z}_{84} to mathbb {Z}_{7}$ defined by $phi(x+84mathbb {Z}):=x+7mathbb {Z}$ and found that $ker(phi) = (7)$ and said that $phi$ is clearly surjective. Is this thinking correct?
Moving onto the other two isomorphism theorems I'm having trouble with these.
For the second theorem, I think that using the homomorphism $phi(x+7mathbb {Z}):=x+7mathbb {Z}$ and $S = mathbb{Z}/84mathbb{Z}$ and $I = 7mathbb{Z}$ might work, but I'm having trouble showing $S+I=mathbb{Z}$ and $S ∩ I = 7mathbb{Z}$
For the third theorem, I think that using the homomorphism $phi(x+7mathbb {Z}):=x+84mathbb {Z} + 7mathbb {Z} $ and $R = mathbb{Z} $ and $I=84mathbb{Z}$ and $J = 7mathbb{Z}$ might work, but I'm having trouble showing $J/I=7mathbb{Z}$
Also would a general form of this $mathbb {Z}_{m}/(n) cong mathbb {Z}_{n}$ if $n|m$?
abstract-algebra ring-theory
asked Dec 10 '18 at 23:27
MacMac
363
$endgroup$
add a comment |
$begingroup$
Prove $mathbb {Z}_{84}/(7) cong mathbb {Z}_{7}$ using each of the three isomorphism theorems for rings.
For the first isomorphism theorem I defined a homomorphism $phi: mathbb {Z}_{84} to mathbb {Z}_{7}$ defined by $phi(x+84mathbb {Z}):=x+7mathbb {Z}$ and found that $ker(phi) = (7)$ and said that $phi$ is clearly surjective. Is this thinking correct?
Moving onto the other two isomorphism theorems I'm having trouble with these.
For the second theorem, I think that using the homomorphism $phi(x+7mathbb {Z}):=x+7mathbb {Z}$ and $S = mathbb{Z}/84mathbb{Z}$ and $I = 7mathbb{Z}$ might work, but I'm having trouble showing $S+I=mathbb{Z}$ and $S ∩ I = 7mathbb{Z}$
For the third theorem, I think that using the homomorphism $phi(x+7mathbb {Z}):=x+84mathbb {Z} + 7mathbb {Z} $ and $R = mathbb{Z} $ and $I=84mathbb{Z}$ and $J = 7mathbb{Z}$ might work, but I'm having trouble showing $J/I=7mathbb{Z}$
Also would a general form of this $mathbb {Z}_{m}/(n) cong mathbb {Z}_{n}$ if $n|m$?
abstract-algebra ring-theory
asked Dec 10 '18 at 23:27
MacMac
363
$endgroup$
add a comment |
$begingroup$
Prove $mathbb {Z}_{84}/(7) cong mathbb {Z}_{7}$ using each of the three isomorphism theorems for rings.
For the first isomorphism theorem I defined a homomorphism $phi: mathbb {Z}_{84} to mathbb {Z}_{7}$ defined by $phi(x+84mathbb {Z}):=x+7mathbb {Z}$ and found that $ker(phi) = (7)$ and said that $phi$ is clearly surjective. Is this thinking correct?
Moving onto the other two isomorphism theorems I'm having trouble with these.
For the second theorem, I think that using the homomorphism $phi(x+7mathbb {Z}):=x+7mathbb {Z}$ and $S = mathbb{Z}/84mathbb{Z}$ and $I = 7mathbb{Z}$ might work, but I'm having trouble showing $S+I=mathbb{Z}$ and $S ∩ I = 7mathbb{Z}$
For the third theorem, I think that using the homomorphism $phi(x+7mathbb {Z}):=x+84mathbb {Z} + 7mathbb {Z} $ and $R = mathbb{Z} $ and $I=84mathbb{Z}$ and $J = 7mathbb{Z}$ might work, but I'm having trouble showing $J/I=7mathbb{Z}$
Also would a general form of this $mathbb {Z}_{m}/(n) cong mathbb {Z}_{n}$ if $n|m$?
abstract-algebra ring-theory
asked Dec 10 '18 at 23:27
MacMac
363
$endgroup$
Prove $mathbb {Z}_{84}/(7) cong mathbb {Z}_{7}$ using each of the three isomorphism theorems for rings.
For the first isomorphism theorem I defined a homomorphism $phi: mathbb {Z}_{84} to mathbb {Z}_{7}$ defined by $phi(x+84mathbb {Z}):=x+7mathbb {Z}$ and found that $ker(phi) = (7)$ and said that $phi$ is clearly surjective. Is this thinking correct?
Moving onto the other two isomorphism theorems I'm having trouble with these.
For the second theorem, I think that using the homomorphism $phi(x+7mathbb {Z}):=x+7mathbb {Z}$ and $S = mathbb{Z}/84mathbb{Z}$ and $I = 7mathbb{Z}$ might work, but I'm having trouble showing $S+I=mathbb{Z}$ and $S ∩ I = 7mathbb{Z}$
For the third theorem, I think that using the homomorphism $phi(x+7mathbb {Z}):=x+84mathbb {Z} + 7mathbb {Z} $ and $R = mathbb{Z} $ and $I=84mathbb{Z}$ and $J = 7mathbb{Z}$ might work, but I'm having trouble showing $J/I=7mathbb{Z}$
Also would a general form of this $mathbb {Z}_{m}/(n) cong mathbb {Z}_{n}$ if $n|m$?
abstract-algebra ring-theory
abstract-algebra ring-theory
asked Dec 10 '18 at 23:27
MacMac
363
asked Dec 10 '18 at 23:27
MacMac
363
asked Dec 10 '18 at 23:27
MacMac
363
asked Dec 10 '18 at 23:27
MacMac
363
asked Dec 10 '18 at 23:27
MacMac
363
363
add a comment |
add a comment |
1 Answer
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$begingroup$
Your answer for the first isomorphism theorem is correct, but you would have to check that $phi$ is well-defined upto choice of representative.
For the second isomorphism theorem, $S$ must be a subring of $mathbb{Z}$; $mathbb{Z}/84mathbb{Z}$ is not a subring of $mathbb{Z}$.
For the third theorem, $J/I$ is an ideal of $mathbb{Z}/84mathbb{Z}$ not $mathbb{Z}$, and in fact $J/I = 7mathbb{Z}/84mathbb{Z} simeq 7(mathbb{Z}/84mathbb{Z})$. The result is then an immediate application of the third isomorphism theorem.
Make sure you have all your rings and ideals straight!
answered Dec 11 '18 at 0:39
Aniruddh AgarwalAniruddh Agarwal
1218
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
Your answer for the first isomorphism theorem is correct, but you would have to check that $phi$ is well-defined upto choice of representative.
For the second isomorphism theorem, $S$ must be a subring of $mathbb{Z}$; $mathbb{Z}/84mathbb{Z}$ is not a subring of $mathbb{Z}$.
For the third theorem, $J/I$ is an ideal of $mathbb{Z}/84mathbb{Z}$ not $mathbb{Z}$, and in fact $J/I = 7mathbb{Z}/84mathbb{Z} simeq 7(mathbb{Z}/84mathbb{Z})$. The result is then an immediate application of the third isomorphism theorem.
Make sure you have all your rings and ideals straight!
answered Dec 11 '18 at 0:39
Aniruddh AgarwalAniruddh Agarwal
1218
$endgroup$
add a comment |
$begingroup$
Your answer for the first isomorphism theorem is correct, but you would have to check that $phi$ is well-defined upto choice of representative.
For the second isomorphism theorem, $S$ must be a subring of $mathbb{Z}$; $mathbb{Z}/84mathbb{Z}$ is not a subring of $mathbb{Z}$.
For the third theorem, $J/I$ is an ideal of $mathbb{Z}/84mathbb{Z}$ not $mathbb{Z}$, and in fact $J/I = 7mathbb{Z}/84mathbb{Z} simeq 7(mathbb{Z}/84mathbb{Z})$. The result is then an immediate application of the third isomorphism theorem.
Make sure you have all your rings and ideals straight!
answered Dec 11 '18 at 0:39
Aniruddh AgarwalAniruddh Agarwal
1218
$endgroup$
add a comment |
$begingroup$
Your answer for the first isomorphism theorem is correct, but you would have to check that $phi$ is well-defined upto choice of representative.
For the second isomorphism theorem, $S$ must be a subring of $mathbb{Z}$; $mathbb{Z}/84mathbb{Z}$ is not a subring of $mathbb{Z}$.
For the third theorem, $J/I$ is an ideal of $mathbb{Z}/84mathbb{Z}$ not $mathbb{Z}$, and in fact $J/I = 7mathbb{Z}/84mathbb{Z} simeq 7(mathbb{Z}/84mathbb{Z})$. The result is then an immediate application of the third isomorphism theorem.
Make sure you have all your rings and ideals straight!
answered Dec 11 '18 at 0:39
Aniruddh AgarwalAniruddh Agarwal
1218
$endgroup$
Your answer for the first isomorphism theorem is correct, but you would have to check that $phi$ is well-defined upto choice of representative.
For the second isomorphism theorem, $S$ must be a subring of $mathbb{Z}$; $mathbb{Z}/84mathbb{Z}$ is not a subring of $mathbb{Z}$.
For the third theorem, $J/I$ is an ideal of $mathbb{Z}/84mathbb{Z}$ not $mathbb{Z}$, and in fact $J/I = 7mathbb{Z}/84mathbb{Z} simeq 7(mathbb{Z}/84mathbb{Z})$. The result is then an immediate application of the third isomorphism theorem.
Make sure you have all your rings and ideals straight!
answered Dec 11 '18 at 0:39
Aniruddh AgarwalAniruddh Agarwal
1218
answered Dec 11 '18 at 0:39
Aniruddh AgarwalAniruddh Agarwal
1218
answered Dec 11 '18 at 0:39
Aniruddh AgarwalAniruddh Agarwal
1218
answered Dec 11 '18 at 0:39
Aniruddh AgarwalAniruddh Agarwal
1218
1218
add a comment |
add a comment |
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