Why $2mathbbZ $ is not integral domain?

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I know that $mathbbZ$ is an integral domain .if $ca = cb$, where $a, b, c in Z$ and $c neq 0$...

Now my Question is that Why $2mathbbZ $ is not integral domain ?

pliz help me,,,,

thanks u

edited Jul 16 at 9:32

Bernard

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asked Jul 16 at 9:14

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up vote
-2
down vote

favorite

I know that $mathbbZ$ is an integral domain .if $ca = cb$, where $a, b, c in Z$ and $c neq 0$...

Now my Question is that Why $2mathbbZ $ is not integral domain ?

pliz help me,,,,

thanks u

edited Jul 16 at 9:32

Bernard

110k635103

asked Jul 16 at 9:14

stupid

57319

add a commentÂ |Â

up vote
-2
down vote

favorite

I know that $mathbbZ$ is an integral domain .if $ca = cb$, where $a, b, c in Z$ and $c neq 0$...

Now my Question is that Why $2mathbbZ $ is not integral domain ?

pliz help me,,,,

thanks u

edited Jul 16 at 9:32

Bernard

110k635103

asked Jul 16 at 9:14

stupid

57319

I know that $mathbbZ$ is an integral domain .if $ca = cb$, where $a, b, c in Z$ and $c neq 0$...

Now my Question is that Why $2mathbbZ $ is not integral domain ?

pliz help me,,,,

thanks u

edited Jul 16 at 9:32

Bernard

110k635103

asked Jul 16 at 9:14

stupid

57319

edited Jul 16 at 9:32

Bernard

110k635103

edited Jul 16 at 9:32

Bernard

110k635103

edited Jul 16 at 9:32

Bernard

110k635103

asked Jul 16 at 9:14

stupid

57319

asked Jul 16 at 9:14

stupid

57319

asked Jul 16 at 9:14

stupid

57319

add a commentÂ |Â

2 Answers
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In commutative ring theory, we generally require that a ring contains a multiplicative identity element. Such an element is not contained in $2Bbb Z$, so we wouldn't consider it a ring, and therefore not an integral domain.

If your ring theory does not require a multiplicative identity, then $2Bbb Z$ is a ring. In that case, it would also be an integral domain.

answered Jul 16 at 9:18

Arthur

98.9k793175

Hmm I'm confused... why are there so many definitions for rings/integral domains? In Dummit & Foote where I learned, $2mathbbZ$ would be a ring and not an integral domain, because it doesn't have an identity. If someone is talking about rings, how can I know which definition they're using?
â€“Â Ovi
Jul 16 at 9:35

2

@Ovi If they don't tell you, you can't know which definition they're using. That's just the way it is, sorry. Some times you may be able to glean from context, but that's it. As for why there are so many definitions, that's just what happens when a field is developed independently by many mathematicians, none of whom know what will actually turn out to be important in the end.
â€“Â Arthur
Jul 16 at 9:36

Oh I thought there may be a standard definition which the majority of people use, because everything I've read on stackexcahgen is consistent with the definitions I got from Dummit & Foote
â€“Â Ovi
Jul 16 at 9:38

add a commentÂ |Â

up vote
1
down vote

It depends if you want to have an identity - if you insist on having then it isn't, otherwise it is.

Here's a useful link: https://www.quora.com/Is-2Z-an-integral-domain

answered Jul 16 at 9:18

asdf

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

up vote
2
down vote

If your ring theory does not require a multiplicative identity, then $2Bbb Z$ is a ring. In that case, it would also be an integral domain.

answered Jul 16 at 9:18

Arthur

98.9k793175

Hmm I'm confused... why are there so many definitions for rings/integral domains? In Dummit & Foote where I learned, $2mathbbZ$ would be a ring and not an integral domain, because it doesn't have an identity. If someone is talking about rings, how can I know which definition they're using?
â€“Â Ovi
Jul 16 at 9:35

2

@Ovi If they don't tell you, you can't know which definition they're using. That's just the way it is, sorry. Some times you may be able to glean from context, but that's it. As for why there are so many definitions, that's just what happens when a field is developed independently by many mathematicians, none of whom know what will actually turn out to be important in the end.
â€“Â Arthur
Jul 16 at 9:36

Oh I thought there may be a standard definition which the majority of people use, because everything I've read on stackexcahgen is consistent with the definitions I got from Dummit & Foote
â€“Â Ovi
Jul 16 at 9:38

add a commentÂ |Â

up vote
2
down vote

If your ring theory does not require a multiplicative identity, then $2Bbb Z$ is a ring. In that case, it would also be an integral domain.

answered Jul 16 at 9:18

Arthur

98.9k793175

Hmm I'm confused... why are there so many definitions for rings/integral domains? In Dummit & Foote where I learned, $2mathbbZ$ would be a ring and not an integral domain, because it doesn't have an identity. If someone is talking about rings, how can I know which definition they're using?
â€“Â Ovi
Jul 16 at 9:35

2

@Ovi If they don't tell you, you can't know which definition they're using. That's just the way it is, sorry. Some times you may be able to glean from context, but that's it. As for why there are so many definitions, that's just what happens when a field is developed independently by many mathematicians, none of whom know what will actually turn out to be important in the end.
â€“Â Arthur
Jul 16 at 9:36

Oh I thought there may be a standard definition which the majority of people use, because everything I've read on stackexcahgen is consistent with the definitions I got from Dummit & Foote
â€“Â Ovi
Jul 16 at 9:38

add a commentÂ |Â

up vote
2
down vote

If your ring theory does not require a multiplicative identity, then $2Bbb Z$ is a ring. In that case, it would also be an integral domain.

answered Jul 16 at 9:18

Arthur

98.9k793175

If your ring theory does not require a multiplicative identity, then $2Bbb Z$ is a ring. In that case, it would also be an integral domain.

answered Jul 16 at 9:18

Arthur

98.9k793175

answered Jul 16 at 9:18

Arthur

98.9k793175

answered Jul 16 at 9:18

Arthur

98.9k793175

answered Jul 16 at 9:18

Arthur

98.9k793175

Hmm I'm confused... why are there so many definitions for rings/integral domains? In Dummit & Foote where I learned, $2mathbbZ$ would be a ring and not an integral domain, because it doesn't have an identity. If someone is talking about rings, how can I know which definition they're using?
â€“Â Ovi
Jul 16 at 9:35

2

@Ovi If they don't tell you, you can't know which definition they're using. That's just the way it is, sorry. Some times you may be able to glean from context, but that's it. As for why there are so many definitions, that's just what happens when a field is developed independently by many mathematicians, none of whom know what will actually turn out to be important in the end.
â€“Â Arthur
Jul 16 at 9:36

Oh I thought there may be a standard definition which the majority of people use, because everything I've read on stackexcahgen is consistent with the definitions I got from Dummit & Foote
â€“Â Ovi
Jul 16 at 9:38

add a commentÂ |Â

Hmm I'm confused... why are there so many definitions for rings/integral domains? In Dummit & Foote where I learned, $2mathbbZ$ would be a ring and not an integral domain, because it doesn't have an identity. If someone is talking about rings, how can I know which definition they're using?
â€“Â Ovi
Jul 16 at 9:35

2

@Ovi If they don't tell you, you can't know which definition they're using. That's just the way it is, sorry. Some times you may be able to glean from context, but that's it. As for why there are so many definitions, that's just what happens when a field is developed independently by many mathematicians, none of whom know what will actually turn out to be important in the end.
â€“Â Arthur
Jul 16 at 9:36

Oh I thought there may be a standard definition which the majority of people use, because everything I've read on stackexcahgen is consistent with the definitions I got from Dummit & Foote
â€“Â Ovi
Jul 16 at 9:38

Hmm I'm confused... why are there so many definitions for rings/integral domains? In Dummit & Foote where I learned, $2mathbbZ$ would be a ring and not an integral domain, because it doesn't have an identity. If someone is talking about rings, how can I know which definition they're using?
â€“Â Ovi
Jul 16 at 9:35

@Ovi If they don't tell you, you can't know which definition they're using. That's just the way it is, sorry. Some times you may be able to glean from context, but that's it. As for why there are so many definitions, that's just what happens when a field is developed independently by many mathematicians, none of whom know what will actually turn out to be important in the end.
â€“Â Arthur
Jul 16 at 9:36

Oh I thought there may be a standard definition which the majority of people use, because everything I've read on stackexcahgen is consistent with the definitions I got from Dummit & Foote
â€“Â Ovi
Jul 16 at 9:38

add a commentÂ |Â

up vote
1
down vote

It depends if you want to have an identity - if you insist on having then it isn't, otherwise it is.

Here's a useful link: https://www.quora.com/Is-2Z-an-integral-domain

answered Jul 16 at 9:18

asdf

3,378519

add a commentÂ |Â

up vote
1
down vote

It depends if you want to have an identity - if you insist on having then it isn't, otherwise it is.

Here's a useful link: https://www.quora.com/Is-2Z-an-integral-domain

answered Jul 16 at 9:18

asdf

3,378519

add a commentÂ |Â

up vote
1
down vote

It depends if you want to have an identity - if you insist on having then it isn't, otherwise it is.

Here's a useful link: https://www.quora.com/Is-2Z-an-integral-domain

answered Jul 16 at 9:18

asdf

3,378519

It depends if you want to have an identity - if you insist on having then it isn't, otherwise it is.

Here's a useful link: https://www.quora.com/Is-2Z-an-integral-domain

answered Jul 16 at 9:18

asdf

3,378519

answered Jul 16 at 9:18

asdf

3,378519

answered Jul 16 at 9:18

asdf

3,378519

answered Jul 16 at 9:18

asdf

3,378519

add a commentÂ |Â

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