Maximal Ideal and its relationship with the ring
1
I understand that if $I$ is a maximal ideal of $R$, then $I+xR =R$ $forall x notin I$, but I fail to grasp how the converse can be true. Would anyone be so kind as to offer me a proof/ some hints at how the proof can be done?
abstract-algebra ring-theory maximal-and-prime-ideals
edited Nov 29 '18 at 12:12
MotylaNogaTomkaMazura
6,542917
asked Nov 29 '18 at 12:07
hephaes
1729
add a comment |
1
I understand that if $I$ is a maximal ideal of $R$, then $I+xR =R$ $forall x notin I$, but I fail to grasp how the converse can be true. Would anyone be so kind as to offer me a proof/ some hints at how the proof can be done?
abstract-algebra ring-theory maximal-and-prime-ideals
edited Nov 29 '18 at 12:12
MotylaNogaTomkaMazura
6,542917
asked Nov 29 '18 at 12:07
hephaes
1729
That is not true if $x in I$ !
– barto
Nov 29 '18 at 12:10
2
What do you mean exactly with ‘the converse’?
– Bernard
Nov 29 '18 at 12:14
add a comment |
1
1
1
I understand that if $I$ is a maximal ideal of $R$, then $I+xR =R$ $forall x notin I$, but I fail to grasp how the converse can be true. Would anyone be so kind as to offer me a proof/ some hints at how the proof can be done?
abstract-algebra ring-theory maximal-and-prime-ideals
edited Nov 29 '18 at 12:12
MotylaNogaTomkaMazura
6,542917
asked Nov 29 '18 at 12:07
hephaes
1729
I understand that if $I$ is a maximal ideal of $R$, then $I+xR =R$ $forall x notin I$, but I fail to grasp how the converse can be true. Would anyone be so kind as to offer me a proof/ some hints at how the proof can be done?
abstract-algebra ring-theory maximal-and-prime-ideals
abstract-algebra ring-theory maximal-and-prime-ideals
edited Nov 29 '18 at 12:12
MotylaNogaTomkaMazura
6,542917
asked Nov 29 '18 at 12:07
hephaes
1729
edited Nov 29 '18 at 12:12
MotylaNogaTomkaMazura
6,542917
asked Nov 29 '18 at 12:07
hephaes
1729
edited Nov 29 '18 at 12:12
MotylaNogaTomkaMazura
6,542917
edited Nov 29 '18 at 12:12
MotylaNogaTomkaMazura
6,542917
edited Nov 29 '18 at 12:12
MotylaNogaTomkaMazura
6,542917
6,542917
asked Nov 29 '18 at 12:07
hephaes
1729
asked Nov 29 '18 at 12:07
hephaes
1729
asked Nov 29 '18 at 12:07
hephaes
1729
1729
That is not true if $x in I$ !
– barto
Nov 29 '18 at 12:10
2
What do you mean exactly with ‘the converse’?
– Bernard
Nov 29 '18 at 12:14
add a comment |
That is not true if $x in I$ !
– barto
Nov 29 '18 at 12:10
2
What do you mean exactly with ‘the converse’?
– Bernard
Nov 29 '18 at 12:14
That is not true if $x in I$ !
– barto
Nov 29 '18 at 12:10
That is not true if $x in I$ !
– barto
Nov 29 '18 at 12:10
2
2
What do you mean exactly with ‘the converse’?
– Bernard
Nov 29 '18 at 12:14
What do you mean exactly with ‘the converse’?
– Bernard
Nov 29 '18 at 12:14
add a comment |
1 Answer
1
active
oldest
votes
3
The converse is obviously true. Since if $I$ satisfies the condition then $I$ cannot be contained in any proper ideal $J$ of $R$ because if $xin Jsetminus I$ then $R=Rx +Isubseteq J+Isubseteq J+J =J$.
edited Nov 29 '18 at 18:30
rschwieb
105k1299244
answered Nov 29 '18 at 12:16
MotylaNogaTomkaMazura
6,542917
1
Why the downvote? This is the obvious idea.
– rschwieb
Nov 29 '18 at 12:37
@rschwieb While this is the answer, I disagree that it is “obviously true” — it was certainly not obvious to the OP.
– Santana Afton
Nov 29 '18 at 12:46
2
+1 This is the very obvious idea, and I agree that it is obviously true.
– Servaes
Nov 29 '18 at 13:49
@SantanaAfton If you happened to assign a downvote for that reason, I'd just urge you to reconsider, since the sentiment (although you disagree with it) does not affect the correctness, quality or usefulness of the answer. Downvotes simply because one "does not like the way someone says something" are not very helpful to the community. Although, it would be completely natural to leave a comment opining that the answer would be better without the "obviously." :)
– rschwieb
Nov 29 '18 at 14:08
@rschwieb I voted not because I don’t like the “obvious” sentiment, but because I find such answers as less useful and of lower quality. That said, I see your point. Were the answer edited, I would certainly revoke my downvote and keep my comment up.
– Santana Afton
Nov 29 '18 at 17:57
|
show 1 more comment
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
3
The converse is obviously true. Since if $I$ satisfies the condition then $I$ cannot be contained in any proper ideal $J$ of $R$ because if $xin Jsetminus I$ then $R=Rx +Isubseteq J+Isubseteq J+J =J$.
edited Nov 29 '18 at 18:30
rschwieb
105k1299244
answered Nov 29 '18 at 12:16
MotylaNogaTomkaMazura
6,542917
1
Why the downvote? This is the obvious idea.
– rschwieb
Nov 29 '18 at 12:37
@rschwieb While this is the answer, I disagree that it is “obviously true” — it was certainly not obvious to the OP.
– Santana Afton
Nov 29 '18 at 12:46
2
+1 This is the very obvious idea, and I agree that it is obviously true.
– Servaes
Nov 29 '18 at 13:49
@SantanaAfton If you happened to assign a downvote for that reason, I'd just urge you to reconsider, since the sentiment (although you disagree with it) does not affect the correctness, quality or usefulness of the answer. Downvotes simply because one "does not like the way someone says something" are not very helpful to the community. Although, it would be completely natural to leave a comment opining that the answer would be better without the "obviously." :)
– rschwieb
Nov 29 '18 at 14:08
@rschwieb I voted not because I don’t like the “obvious” sentiment, but because I find such answers as less useful and of lower quality. That said, I see your point. Were the answer edited, I would certainly revoke my downvote and keep my comment up.
– Santana Afton
Nov 29 '18 at 17:57
|
show 1 more comment
3
The converse is obviously true. Since if $I$ satisfies the condition then $I$ cannot be contained in any proper ideal $J$ of $R$ because if $xin Jsetminus I$ then $R=Rx +Isubseteq J+Isubseteq J+J =J$.
edited Nov 29 '18 at 18:30
rschwieb
105k1299244
answered Nov 29 '18 at 12:16
MotylaNogaTomkaMazura
6,542917
1
Why the downvote? This is the obvious idea.
– rschwieb
Nov 29 '18 at 12:37
@rschwieb While this is the answer, I disagree that it is “obviously true” — it was certainly not obvious to the OP.
– Santana Afton
Nov 29 '18 at 12:46
2
+1 This is the very obvious idea, and I agree that it is obviously true.
– Servaes
Nov 29 '18 at 13:49
@SantanaAfton If you happened to assign a downvote for that reason, I'd just urge you to reconsider, since the sentiment (although you disagree with it) does not affect the correctness, quality or usefulness of the answer. Downvotes simply because one "does not like the way someone says something" are not very helpful to the community. Although, it would be completely natural to leave a comment opining that the answer would be better without the "obviously." :)
– rschwieb
Nov 29 '18 at 14:08
@rschwieb I voted not because I don’t like the “obvious” sentiment, but because I find such answers as less useful and of lower quality. That said, I see your point. Were the answer edited, I would certainly revoke my downvote and keep my comment up.
– Santana Afton
Nov 29 '18 at 17:57
|
show 1 more comment
3
3
3
The converse is obviously true. Since if $I$ satisfies the condition then $I$ cannot be contained in any proper ideal $J$ of $R$ because if $xin Jsetminus I$ then $R=Rx +Isubseteq J+Isubseteq J+J =J$.
edited Nov 29 '18 at 18:30
rschwieb
105k1299244
answered Nov 29 '18 at 12:16
MotylaNogaTomkaMazura
6,542917
The converse is obviously true. Since if $I$ satisfies the condition then $I$ cannot be contained in any proper ideal $J$ of $R$ because if $xin Jsetminus I$ then $R=Rx +Isubseteq J+Isubseteq J+J =J$.
edited Nov 29 '18 at 18:30
rschwieb
105k1299244
answered Nov 29 '18 at 12:16
MotylaNogaTomkaMazura
6,542917
edited Nov 29 '18 at 18:30
rschwieb
105k1299244
edited Nov 29 '18 at 18:30
rschwieb
105k1299244
edited Nov 29 '18 at 18:30
rschwieb
105k1299244
105k1299244
answered Nov 29 '18 at 12:16
MotylaNogaTomkaMazura
6,542917
answered Nov 29 '18 at 12:16
MotylaNogaTomkaMazura
6,542917
answered Nov 29 '18 at 12:16
MotylaNogaTomkaMazura
6,542917
6,542917
1
Why the downvote? This is the obvious idea.
– rschwieb
Nov 29 '18 at 12:37
@rschwieb While this is the answer, I disagree that it is “obviously true” — it was certainly not obvious to the OP.
– Santana Afton
Nov 29 '18 at 12:46
2
+1 This is the very obvious idea, and I agree that it is obviously true.
– Servaes
Nov 29 '18 at 13:49
@SantanaAfton If you happened to assign a downvote for that reason, I'd just urge you to reconsider, since the sentiment (although you disagree with it) does not affect the correctness, quality or usefulness of the answer. Downvotes simply because one "does not like the way someone says something" are not very helpful to the community. Although, it would be completely natural to leave a comment opining that the answer would be better without the "obviously." :)
– rschwieb
Nov 29 '18 at 14:08
@rschwieb I voted not because I don’t like the “obvious” sentiment, but because I find such answers as less useful and of lower quality. That said, I see your point. Were the answer edited, I would certainly revoke my downvote and keep my comment up.
– Santana Afton
Nov 29 '18 at 17:57
|
show 1 more comment
1
Why the downvote? This is the obvious idea.
– rschwieb
Nov 29 '18 at 12:37
@rschwieb While this is the answer, I disagree that it is “obviously true” — it was certainly not obvious to the OP.
– Santana Afton
Nov 29 '18 at 12:46
2
+1 This is the very obvious idea, and I agree that it is obviously true.
– Servaes
Nov 29 '18 at 13:49
@SantanaAfton If you happened to assign a downvote for that reason, I'd just urge you to reconsider, since the sentiment (although you disagree with it) does not affect the correctness, quality or usefulness of the answer. Downvotes simply because one "does not like the way someone says something" are not very helpful to the community. Although, it would be completely natural to leave a comment opining that the answer would be better without the "obviously." :)
– rschwieb
Nov 29 '18 at 14:08
@rschwieb I voted not because I don’t like the “obvious” sentiment, but because I find such answers as less useful and of lower quality. That said, I see your point. Were the answer edited, I would certainly revoke my downvote and keep my comment up.
– Santana Afton
Nov 29 '18 at 17:57
1
1
Why the downvote? This is the obvious idea.
– rschwieb
Nov 29 '18 at 12:37
Why the downvote? This is the obvious idea.
– rschwieb
Nov 29 '18 at 12:37
@rschwieb While this is the answer, I disagree that it is “obviously true” — it was certainly not obvious to the OP.
– Santana Afton
Nov 29 '18 at 12:46
@rschwieb While this is the answer, I disagree that it is “obviously true” — it was certainly not obvious to the OP.
– Santana Afton
Nov 29 '18 at 12:46
2
2
+1 This is the very obvious idea, and I agree that it is obviously true.
– Servaes
Nov 29 '18 at 13:49
+1 This is the very obvious idea, and I agree that it is obviously true.
– Servaes
Nov 29 '18 at 13:49
@SantanaAfton If you happened to assign a downvote for that reason, I'd just urge you to reconsider, since the sentiment (although you disagree with it) does not affect the correctness, quality or usefulness of the answer. Downvotes simply because one "does not like the way someone says something" are not very helpful to the community. Although, it would be completely natural to leave a comment opining that the answer would be better without the "obviously." :)
– rschwieb
Nov 29 '18 at 14:08
@SantanaAfton If you happened to assign a downvote for that reason, I'd just urge you to reconsider, since the sentiment (although you disagree with it) does not affect the correctness, quality or usefulness of the answer. Downvotes simply because one "does not like the way someone says something" are not very helpful to the community. Although, it would be completely natural to leave a comment opining that the answer would be better without the "obviously." :)
– rschwieb
Nov 29 '18 at 14:08
@rschwieb I voted not because I don’t like the “obvious” sentiment, but because I find such answers as less useful and of lower quality. That said, I see your point. Were the answer edited, I would certainly revoke my downvote and keep my comment up.
– Santana Afton
Nov 29 '18 at 17:57
@rschwieb I voted not because I don’t like the “obvious” sentiment, but because I find such answers as less useful and of lower quality. That said, I see your point. Were the answer edited, I would certainly revoke my downvote and keep my comment up.
– Santana Afton
Nov 29 '18 at 17:57
|
show 1 more comment
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That is not true if $x in I$ !
– barto
Nov 29 '18 at 12:10
2
What do you mean exactly with ‘the converse’?
– Bernard
Nov 29 '18 at 12:14