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Why Krull dimension of zero ring defined to be negative or it is just a convention?
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From wikipedia I have accrossed this claim "The Krull dimension of the zero ring is typically defined to be either $displaystyle -infty $ or $displaystyle -1 $. The zero ring is the only ring with a negative dimension " . Now my question here is :Why krull dimension of zero ring defined to be negative or just a convention ?
ring-theory convention krull-dimension
edited Jul 21 at 21:50
Bernard
110k635103
asked Jul 21 at 20:34
zeraoulia rafik
2,1071823
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From wikipedia I have accrossed this claim "The Krull dimension of the zero ring is typically defined to be either $displaystyle -infty $ or $displaystyle -1 $. The zero ring is the only ring with a negative dimension " . Now my question here is :Why krull dimension of zero ring defined to be negative or just a convention ?
ring-theory convention krull-dimension
edited Jul 21 at 21:50
Bernard
110k635103
asked Jul 21 at 20:34
zeraoulia rafik
2,1071823
1
It should be $-infty$. You want Krull dimension to be additive with respect to, say, products of varieties, and the product of the empty variety (the spectrum of the zero ring) with any variety is the empty variety.
– Qiaochu Yuan
Jul 21 at 23:04
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
From wikipedia I have accrossed this claim "The Krull dimension of the zero ring is typically defined to be either $displaystyle -infty $ or $displaystyle -1 $. The zero ring is the only ring with a negative dimension " . Now my question here is :Why krull dimension of zero ring defined to be negative or just a convention ?
ring-theory convention krull-dimension
edited Jul 21 at 21:50
Bernard
110k635103
asked Jul 21 at 20:34
zeraoulia rafik
2,1071823
From wikipedia I have accrossed this claim "The Krull dimension of the zero ring is typically defined to be either $displaystyle -infty $ or $displaystyle -1 $. The zero ring is the only ring with a negative dimension " . Now my question here is :Why krull dimension of zero ring defined to be negative or just a convention ?
ring-theory convention krull-dimension
edited Jul 21 at 21:50
Bernard
110k635103
asked Jul 21 at 20:34
zeraoulia rafik
2,1071823
edited Jul 21 at 21:50
Bernard
110k635103
edited Jul 21 at 21:50
Bernard
110k635103
edited Jul 21 at 21:50
Bernard
110k635103
110k635103
asked Jul 21 at 20:34
zeraoulia rafik
2,1071823
asked Jul 21 at 20:34
zeraoulia rafik
2,1071823
asked Jul 21 at 20:34
zeraoulia rafik
2,1071823
2,1071823
1
It should be $-infty$. You want Krull dimension to be additive with respect to, say, products of varieties, and the product of the empty variety (the spectrum of the zero ring) with any variety is the empty variety.
– Qiaochu Yuan
Jul 21 at 23:04
add a comment |Â
1
It should be $-infty$. You want Krull dimension to be additive with respect to, say, products of varieties, and the product of the empty variety (the spectrum of the zero ring) with any variety is the empty variety.
– Qiaochu Yuan
Jul 21 at 23:04
1
1
It should be $-infty$. You want Krull dimension to be additive with respect to, say, products of varieties, and the product of the empty variety (the spectrum of the zero ring) with any variety is the empty variety.
– Qiaochu Yuan
Jul 21 at 23:04
It should be $-infty$. You want Krull dimension to be additive with respect to, say, products of varieties, and the product of the empty variety (the spectrum of the zero ring) with any variety is the empty variety.
– Qiaochu Yuan
Jul 21 at 23:04
add a comment |Â
1 Answer
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The least upper bound of the length of a chain of prime ideals in the zero ring is $-infty$, since the zero ring has no prime ideals.
answered Jul 21 at 20:46
Kevin Carlson
29.2k23065
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
The least upper bound of the length of a chain of prime ideals in the zero ring is $-infty$, since the zero ring has no prime ideals.
answered Jul 21 at 20:46
Kevin Carlson
29.2k23065
add a comment |Â
up vote
3
down vote
The least upper bound of the length of a chain of prime ideals in the zero ring is $-infty$, since the zero ring has no prime ideals.
answered Jul 21 at 20:46
Kevin Carlson
29.2k23065
add a comment |Â
up vote
3
down vote
up vote
3
down vote
The least upper bound of the length of a chain of prime ideals in the zero ring is $-infty$, since the zero ring has no prime ideals.
answered Jul 21 at 20:46
Kevin Carlson
29.2k23065
The least upper bound of the length of a chain of prime ideals in the zero ring is $-infty$, since the zero ring has no prime ideals.
answered Jul 21 at 20:46
Kevin Carlson
29.2k23065
answered Jul 21 at 20:46
Kevin Carlson
29.2k23065
answered Jul 21 at 20:46
Kevin Carlson
29.2k23065
answered Jul 21 at 20:46
Kevin Carlson
29.2k23065
29.2k23065
add a comment |Â
add a comment |Â
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1
It should be $-infty$. You want Krull dimension to be additive with respect to, say, products of varieties, and the product of the empty variety (the spectrum of the zero ring) with any variety is the empty variety.
– Qiaochu Yuan
Jul 21 at 23:04