Question about the zero ideal in $mathbb{Z}$












0












$begingroup$


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







share|cite|improve this question









$endgroup$








  • 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




    $begingroup$
    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


















0












$begingroup$


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







share|cite|improve this question









$endgroup$








  • 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




    $begingroup$
    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
















0












0








0





$begingroup$


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







share|cite|improve this question









$endgroup$




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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










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
















  • 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












1 Answer
1






active

oldest

votes


















2












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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












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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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
















2












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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














2












2








2





$begingroup$

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.






share|cite|improve this answer









$endgroup$



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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










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


















  • $begingroup$
    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


















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