Non-zero Ideal in an Integral Domain is indecomposable
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Need to prove that : any non-zero ideal in an integral domain is indecomposable.
Now if $I=Abigoplus B$, then $A$ and $B$ are subgroups of $I$, if I am not wrong. Does it say anything about having idempotents, then in that case multiplication of the two idempotents will be zero leading to a contradiction that we are in a domain. Please help.
abstract-algebra ring-theory ideals integral-domain
asked Nov 17 at 17:12
PSG
1999
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up vote
1
down vote
favorite
Need to prove that : any non-zero ideal in an integral domain is indecomposable.
Now if $I=Abigoplus B$, then $A$ and $B$ are subgroups of $I$, if I am not wrong. Does it say anything about having idempotents, then in that case multiplication of the two idempotents will be zero leading to a contradiction that we are in a domain. Please help.
abstract-algebra ring-theory ideals integral-domain
asked Nov 17 at 17:12
PSG
1999
The decomposition of an ideal doesn’t always have anything to do with idempotents. In a domain, there are only trivial idempotents. There are noncommutative domains in which there are right ideals with infinitely many nonzero summands.
– rschwieb
Nov 17 at 19:22
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Need to prove that : any non-zero ideal in an integral domain is indecomposable.
Now if $I=Abigoplus B$, then $A$ and $B$ are subgroups of $I$, if I am not wrong. Does it say anything about having idempotents, then in that case multiplication of the two idempotents will be zero leading to a contradiction that we are in a domain. Please help.
abstract-algebra ring-theory ideals integral-domain
asked Nov 17 at 17:12
PSG
1999
Need to prove that : any non-zero ideal in an integral domain is indecomposable.
Now if $I=Abigoplus B$, then $A$ and $B$ are subgroups of $I$, if I am not wrong. Does it say anything about having idempotents, then in that case multiplication of the two idempotents will be zero leading to a contradiction that we are in a domain. Please help.
abstract-algebra ring-theory ideals integral-domain
abstract-algebra ring-theory ideals integral-domain
asked Nov 17 at 17:12
PSG
1999
asked Nov 17 at 17:12
PSG
1999
edited Nov 17 at 17:48
asked Nov 17 at 17:12
PSG
1999
asked Nov 17 at 17:12
PSG
1999
asked Nov 17 at 17:12
PSG
1999
1999
The decomposition of an ideal doesn’t always have anything to do with idempotents. In a domain, there are only trivial idempotents. There are noncommutative domains in which there are right ideals with infinitely many nonzero summands.
– rschwieb
Nov 17 at 19:22
add a comment |
The decomposition of an ideal doesn’t always have anything to do with idempotents. In a domain, there are only trivial idempotents. There are noncommutative domains in which there are right ideals with infinitely many nonzero summands.
– rschwieb
Nov 17 at 19:22
The decomposition of an ideal doesn’t always have anything to do with idempotents. In a domain, there are only trivial idempotents. There are noncommutative domains in which there are right ideals with infinitely many nonzero summands.
– rschwieb
Nov 17 at 19:22
The decomposition of an ideal doesn’t always have anything to do with idempotents. In a domain, there are only trivial idempotents. There are noncommutative domains in which there are right ideals with infinitely many nonzero summands.
– rschwieb
Nov 17 at 19:22
add a comment |
1 Answer
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oldest
votes
up vote
1
down vote
accepted
$Acap Bsupseteq ABneq {0}$ If $A$ and $B$ are nonzero.
answered Nov 17 at 18:37
rschwieb
103k1299238
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
$Acap Bsupseteq ABneq {0}$ If $A$ and $B$ are nonzero.
answered Nov 17 at 18:37
rschwieb
103k1299238
add a comment |
up vote
1
down vote
accepted
$Acap Bsupseteq ABneq {0}$ If $A$ and $B$ are nonzero.
answered Nov 17 at 18:37
rschwieb
103k1299238
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
$Acap Bsupseteq ABneq {0}$ If $A$ and $B$ are nonzero.
answered Nov 17 at 18:37
rschwieb
103k1299238
$Acap Bsupseteq ABneq {0}$ If $A$ and $B$ are nonzero.
answered Nov 17 at 18:37
rschwieb
103k1299238
answered Nov 17 at 18:37
rschwieb
103k1299238
answered Nov 17 at 18:37
rschwieb
103k1299238
answered Nov 17 at 18:37
rschwieb
103k1299238
103k1299238
add a comment |
add a comment |
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The decomposition of an ideal doesn’t always have anything to do with idempotents. In a domain, there are only trivial idempotents. There are noncommutative domains in which there are right ideals with infinitely many nonzero summands.
– rschwieb
Nov 17 at 19:22