Question about the zero ideal in $mathbb{Z}$
$begingroup$
I have been studying Ring theory and a question came up.
Theorem: A commutative ring $R$ is integral domain iff the zero ideal $I={0}$ is prime ideal in $R$.
Question: How can it be that in $mathbb{Z}$ we do not count $I={0}$ as a prime ideal ?(That's what is stated in my book)
We know that $mathbb{Z}$ is integral domain. So the zero ideal must be prime.
abstract-algebra ring-theory
asked Dec 23 '18 at 19:19
argiriskarargiriskar
1409
$endgroup$
add a comment |
$begingroup$
I have been studying Ring theory and a question came up.
Theorem: A commutative ring $R$ is integral domain iff the zero ideal $I={0}$ is prime ideal in $R$.
Question: How can it be that in $mathbb{Z}$ we do not count $I={0}$ as a prime ideal ?(That's what is stated in my book)
We know that $mathbb{Z}$ is integral domain. So the zero ideal must be prime.
abstract-algebra ring-theory
asked Dec 23 '18 at 19:19
argiriskarargiriskar
1409
$endgroup$
2
$begingroup$
$(0)$ is a prime ideal in $Bbb Z$, if $abin (0)$ then either $ain (0)$ or $bin (0)$.
$endgroup$
– cansomeonehelpmeout
Dec 23 '18 at 19:21
3
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Can you show us exactly what your book states?
$endgroup$
– Eric Wofsey
Dec 23 '18 at 19:22
1
$begingroup$
The zero ideal is a prime one iff the (non-trivial) ring is an integral domain. If it is stated otherwise in your book or anywhere then that is wrong
$endgroup$
– DonAntonio
Dec 23 '18 at 19:24
1
$begingroup$
See Why doesn't $0$ being a prime ideal in $mathbb Z$ imply that $0$ is a prime number?. If this is really what you are asking about then we can close as a dupe of that (and likely many others).
$endgroup$
– Bill Dubuque
Dec 23 '18 at 19:36
add a comment |
$begingroup$
I have been studying Ring theory and a question came up.
Theorem: A commutative ring $R$ is integral domain iff the zero ideal $I={0}$ is prime ideal in $R$.
Question: How can it be that in $mathbb{Z}$ we do not count $I={0}$ as a prime ideal ?(That's what is stated in my book)
We know that $mathbb{Z}$ is integral domain. So the zero ideal must be prime.
abstract-algebra ring-theory
asked Dec 23 '18 at 19:19
argiriskarargiriskar
1409
$endgroup$
I have been studying Ring theory and a question came up.
Theorem: A commutative ring $R$ is integral domain iff the zero ideal $I={0}$ is prime ideal in $R$.
Question: How can it be that in $mathbb{Z}$ we do not count $I={0}$ as a prime ideal ?(That's what is stated in my book)
We know that $mathbb{Z}$ is integral domain. So the zero ideal must be prime.
abstract-algebra ring-theory
abstract-algebra ring-theory
asked Dec 23 '18 at 19:19
argiriskarargiriskar
1409
asked Dec 23 '18 at 19:19
argiriskarargiriskar
1409
asked Dec 23 '18 at 19:19
argiriskarargiriskar
1409
asked Dec 23 '18 at 19:19
argiriskarargiriskar
1409
asked Dec 23 '18 at 19:19
argiriskarargiriskar
1409
1409
2
$begingroup$
$(0)$ is a prime ideal in $Bbb Z$, if $abin (0)$ then either $ain (0)$ or $bin (0)$.
$endgroup$
– cansomeonehelpmeout
Dec 23 '18 at 19:21
3
$begingroup$
Can you show us exactly what your book states?
$endgroup$
– Eric Wofsey
Dec 23 '18 at 19:22
1
$begingroup$
The zero ideal is a prime one iff the (non-trivial) ring is an integral domain. If it is stated otherwise in your book or anywhere then that is wrong
$endgroup$
– DonAntonio
Dec 23 '18 at 19:24
1
$begingroup$
See Why doesn't $0$ being a prime ideal in $mathbb Z$ imply that $0$ is a prime number?. If this is really what you are asking about then we can close as a dupe of that (and likely many others).
$endgroup$
– Bill Dubuque
Dec 23 '18 at 19:36
add a comment |
2
$begingroup$
$(0)$ is a prime ideal in $Bbb Z$, if $abin (0)$ then either $ain (0)$ or $bin (0)$.
$endgroup$
– cansomeonehelpmeout
Dec 23 '18 at 19:21
3
$begingroup$
Can you show us exactly what your book states?
$endgroup$
– Eric Wofsey
Dec 23 '18 at 19:22
1
$begingroup$
The zero ideal is a prime one iff the (non-trivial) ring is an integral domain. If it is stated otherwise in your book or anywhere then that is wrong
$endgroup$
– DonAntonio
Dec 23 '18 at 19:24
1
$begingroup$
See Why doesn't $0$ being a prime ideal in $mathbb Z$ imply that $0$ is a prime number?. If this is really what you are asking about then we can close as a dupe of that (and likely many others).
$endgroup$
– Bill Dubuque
Dec 23 '18 at 19:36
2
2
$begingroup$
$(0)$ is a prime ideal in $Bbb Z$, if $abin (0)$ then either $ain (0)$ or $bin (0)$.
$endgroup$
– cansomeonehelpmeout
Dec 23 '18 at 19:21
$begingroup$
$(0)$ is a prime ideal in $Bbb Z$, if $abin (0)$ then either $ain (0)$ or $bin (0)$.
$endgroup$
– cansomeonehelpmeout
Dec 23 '18 at 19:21
3
3
$begingroup$
Can you show us exactly what your book states?
$endgroup$
– Eric Wofsey
Dec 23 '18 at 19:22
$begingroup$
Can you show us exactly what your book states?
$endgroup$
– Eric Wofsey
Dec 23 '18 at 19:22
1
1
$begingroup$
The zero ideal is a prime one iff the (non-trivial) ring is an integral domain. If it is stated otherwise in your book or anywhere then that is wrong
$endgroup$
– DonAntonio
Dec 23 '18 at 19:24
$begingroup$
The zero ideal is a prime one iff the (non-trivial) ring is an integral domain. If it is stated otherwise in your book or anywhere then that is wrong
$endgroup$
– DonAntonio
Dec 23 '18 at 19:24
1
1
$begingroup$
See Why doesn't $0$ being a prime ideal in $mathbb Z$ imply that $0$ is a prime number?. If this is really what you are asking about then we can close as a dupe of that (and likely many others).
$endgroup$
– Bill Dubuque
Dec 23 '18 at 19:36
$begingroup$
See Why doesn't $0$ being a prime ideal in $mathbb Z$ imply that $0$ is a prime number?. If this is really what you are asking about then we can close as a dupe of that (and likely many others).
$endgroup$
– Bill Dubuque
Dec 23 '18 at 19:36
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
An ideal is prime if whenever it contains a product it contains one of the factors. The zero ideal has that property just for integral domains.
answered Dec 23 '18 at 19:22
Ethan BolkerEthan Bolker
46.5k555122
$endgroup$
$begingroup$
So the zero ideal is prime in the integers.
$endgroup$
– argiriskar
Dec 23 '18 at 19:23
3
$begingroup$
$I=0$ is a prime ideal iff $Bbb{Z}/I=Bbb{Z}$ is an integral domain. So yes.
$endgroup$
– Dietrich Burde
Dec 23 '18 at 19:24
$begingroup$
Thank you very much!
$endgroup$
– argiriskar
Dec 23 '18 at 19:25
add a comment |
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1 Answer
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1 Answer
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$begingroup$
An ideal is prime if whenever it contains a product it contains one of the factors. The zero ideal has that property just for integral domains.
answered Dec 23 '18 at 19:22
Ethan BolkerEthan Bolker
46.5k555122
$endgroup$
$begingroup$
So the zero ideal is prime in the integers.
$endgroup$
– argiriskar
Dec 23 '18 at 19:23
3
$begingroup$
$I=0$ is a prime ideal iff $Bbb{Z}/I=Bbb{Z}$ is an integral domain. So yes.
$endgroup$
– Dietrich Burde
Dec 23 '18 at 19:24
$begingroup$
Thank you very much!
$endgroup$
– argiriskar
Dec 23 '18 at 19:25
add a comment |
$begingroup$
An ideal is prime if whenever it contains a product it contains one of the factors. The zero ideal has that property just for integral domains.
answered Dec 23 '18 at 19:22
Ethan BolkerEthan Bolker
46.5k555122
$endgroup$
$begingroup$
So the zero ideal is prime in the integers.
$endgroup$
– argiriskar
Dec 23 '18 at 19:23
3
$begingroup$
$I=0$ is a prime ideal iff $Bbb{Z}/I=Bbb{Z}$ is an integral domain. So yes.
$endgroup$
– Dietrich Burde
Dec 23 '18 at 19:24
$begingroup$
Thank you very much!
$endgroup$
– argiriskar
Dec 23 '18 at 19:25
add a comment |
$begingroup$
An ideal is prime if whenever it contains a product it contains one of the factors. The zero ideal has that property just for integral domains.
answered Dec 23 '18 at 19:22
Ethan BolkerEthan Bolker
46.5k555122
$endgroup$
An ideal is prime if whenever it contains a product it contains one of the factors. The zero ideal has that property just for integral domains.
answered Dec 23 '18 at 19:22
Ethan BolkerEthan Bolker
46.5k555122
answered Dec 23 '18 at 19:22
Ethan BolkerEthan Bolker
46.5k555122
answered Dec 23 '18 at 19:22
Ethan BolkerEthan Bolker
46.5k555122
answered Dec 23 '18 at 19:22
Ethan BolkerEthan Bolker
46.5k555122
46.5k555122
$begingroup$
So the zero ideal is prime in the integers.
$endgroup$
– argiriskar
Dec 23 '18 at 19:23
3
$begingroup$
$I=0$ is a prime ideal iff $Bbb{Z}/I=Bbb{Z}$ is an integral domain. So yes.
$endgroup$
– Dietrich Burde
Dec 23 '18 at 19:24
$begingroup$
Thank you very much!
$endgroup$
– argiriskar
Dec 23 '18 at 19:25
add a comment |
$begingroup$
So the zero ideal is prime in the integers.
$endgroup$
– argiriskar
Dec 23 '18 at 19:23
3
$begingroup$
$I=0$ is a prime ideal iff $Bbb{Z}/I=Bbb{Z}$ is an integral domain. So yes.
$endgroup$
– Dietrich Burde
Dec 23 '18 at 19:24
$begingroup$
Thank you very much!
$endgroup$
– argiriskar
Dec 23 '18 at 19:25
$begingroup$
So the zero ideal is prime in the integers.
$endgroup$
– argiriskar
Dec 23 '18 at 19:23
$begingroup$
So the zero ideal is prime in the integers.
$endgroup$
– argiriskar
Dec 23 '18 at 19:23
3
3
$begingroup$
$I=0$ is a prime ideal iff $Bbb{Z}/I=Bbb{Z}$ is an integral domain. So yes.
$endgroup$
– Dietrich Burde
Dec 23 '18 at 19:24
$begingroup$
$I=0$ is a prime ideal iff $Bbb{Z}/I=Bbb{Z}$ is an integral domain. So yes.
$endgroup$
– Dietrich Burde
Dec 23 '18 at 19:24
$begingroup$
Thank you very much!
$endgroup$
– argiriskar
Dec 23 '18 at 19:25
$begingroup$
Thank you very much!
$endgroup$
– argiriskar
Dec 23 '18 at 19:25
add a comment |
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2
$begingroup$
$(0)$ is a prime ideal in $Bbb Z$, if $abin (0)$ then either $ain (0)$ or $bin (0)$.
$endgroup$
– cansomeonehelpmeout
Dec 23 '18 at 19:21
3
$begingroup$
Can you show us exactly what your book states?
$endgroup$
– Eric Wofsey
Dec 23 '18 at 19:22
1
$begingroup$
The zero ideal is a prime one iff the (non-trivial) ring is an integral domain. If it is stated otherwise in your book or anywhere then that is wrong
$endgroup$
– DonAntonio
Dec 23 '18 at 19:24
1
$begingroup$
See Why doesn't $0$ being a prime ideal in $mathbb Z$ imply that $0$ is a prime number?. If this is really what you are asking about then we can close as a dupe of that (and likely many others).
$endgroup$
– Bill Dubuque
Dec 23 '18 at 19:36