Flat ring homomorphism but not injective.
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Let $Ato B$ be flat ring homomorphism(i.e. $B$ is flat $A$ module.)
If $B$ is faithfully flat, then $Ato B$ is injection.
$textbf{Q:}$ What is the example of flat but not injective ring homomorphism?(i.e. I want to fail faithfully flat but remain flat.) I think I need some ring $B$ as projective which realizes $B=F/N$ where $F$ is free $A-$module and this has to be compatible with ring structure as well. Clearly, I could not get this work over $A$ being a field.
abstract-algebra commutative-algebra
asked Oct 4 at 17:54
user45765
2,4592721
add a comment |
up vote
1
down vote
favorite
Let $Ato B$ be flat ring homomorphism(i.e. $B$ is flat $A$ module.)
If $B$ is faithfully flat, then $Ato B$ is injection.
$textbf{Q:}$ What is the example of flat but not injective ring homomorphism?(i.e. I want to fail faithfully flat but remain flat.) I think I need some ring $B$ as projective which realizes $B=F/N$ where $F$ is free $A-$module and this has to be compatible with ring structure as well. Clearly, I could not get this work over $A$ being a field.
abstract-algebra commutative-algebra
asked Oct 4 at 17:54
user45765
2,4592721
1
$A$ any ring, $B$ the zero ring?
– Lord Shark the Unknown
Oct 4 at 18:01
@LordSharktheUnknown $A,B$ are unital rings.
– user45765
Oct 4 at 18:02
4
The zero ring is unital....
– Lord Shark the Unknown
Oct 4 at 18:03
@LordSharktheUnknown Can I have a unital ring counter example rather than $0$ ring?
– user45765
Oct 4 at 18:05
3
Take $A=Boplus B$ and $Ato B$ the projection onto the first factor.
– David Hill
Oct 4 at 18:05
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $Ato B$ be flat ring homomorphism(i.e. $B$ is flat $A$ module.)
If $B$ is faithfully flat, then $Ato B$ is injection.
$textbf{Q:}$ What is the example of flat but not injective ring homomorphism?(i.e. I want to fail faithfully flat but remain flat.) I think I need some ring $B$ as projective which realizes $B=F/N$ where $F$ is free $A-$module and this has to be compatible with ring structure as well. Clearly, I could not get this work over $A$ being a field.
abstract-algebra commutative-algebra
asked Oct 4 at 17:54
user45765
2,4592721
Let $Ato B$ be flat ring homomorphism(i.e. $B$ is flat $A$ module.)
If $B$ is faithfully flat, then $Ato B$ is injection.
$textbf{Q:}$ What is the example of flat but not injective ring homomorphism?(i.e. I want to fail faithfully flat but remain flat.) I think I need some ring $B$ as projective which realizes $B=F/N$ where $F$ is free $A-$module and this has to be compatible with ring structure as well. Clearly, I could not get this work over $A$ being a field.
abstract-algebra commutative-algebra
abstract-algebra commutative-algebra
asked Oct 4 at 17:54
user45765
2,4592721
asked Oct 4 at 17:54
user45765
2,4592721
asked Oct 4 at 17:54
user45765
2,4592721
asked Oct 4 at 17:54
user45765
2,4592721
asked Oct 4 at 17:54
user45765
2,4592721
2,4592721
1
$A$ any ring, $B$ the zero ring?
– Lord Shark the Unknown
Oct 4 at 18:01
@LordSharktheUnknown $A,B$ are unital rings.
– user45765
Oct 4 at 18:02
4
The zero ring is unital....
– Lord Shark the Unknown
Oct 4 at 18:03
@LordSharktheUnknown Can I have a unital ring counter example rather than $0$ ring?
– user45765
Oct 4 at 18:05
3
Take $A=Boplus B$ and $Ato B$ the projection onto the first factor.
– David Hill
Oct 4 at 18:05
add a comment |
1
$A$ any ring, $B$ the zero ring?
– Lord Shark the Unknown
Oct 4 at 18:01
@LordSharktheUnknown $A,B$ are unital rings.
– user45765
Oct 4 at 18:02
4
The zero ring is unital....
– Lord Shark the Unknown
Oct 4 at 18:03
@LordSharktheUnknown Can I have a unital ring counter example rather than $0$ ring?
– user45765
Oct 4 at 18:05
3
Take $A=Boplus B$ and $Ato B$ the projection onto the first factor.
– David Hill
Oct 4 at 18:05
1
1
$A$ any ring, $B$ the zero ring?
– Lord Shark the Unknown
Oct 4 at 18:01
$A$ any ring, $B$ the zero ring?
– Lord Shark the Unknown
Oct 4 at 18:01
@LordSharktheUnknown $A,B$ are unital rings.
– user45765
Oct 4 at 18:02
@LordSharktheUnknown $A,B$ are unital rings.
– user45765
Oct 4 at 18:02
4
4
The zero ring is unital....
– Lord Shark the Unknown
Oct 4 at 18:03
The zero ring is unital....
– Lord Shark the Unknown
Oct 4 at 18:03
@LordSharktheUnknown Can I have a unital ring counter example rather than $0$ ring?
– user45765
Oct 4 at 18:05
@LordSharktheUnknown Can I have a unital ring counter example rather than $0$ ring?
– user45765
Oct 4 at 18:05
3
3
Take $A=Boplus B$ and $Ato B$ the projection onto the first factor.
– David Hill
Oct 4 at 18:05
Take $A=Boplus B$ and $Ato B$ the projection onto the first factor.
– David Hill
Oct 4 at 18:05
add a comment |
1 Answer
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Let $A$ be a nonzero and absolutely flat ring, let $mathfrak{m}subseteq A$ be a maximal ideal, and let $B=A/mathfrak{m}$. Then, the canonical morphism $Arightarrow B$ is flat but not injective.
answered Nov 22 at 18:13
Fred Rohrer
261110
add a comment |
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1 Answer
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up vote
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down vote
Let $A$ be a nonzero and absolutely flat ring, let $mathfrak{m}subseteq A$ be a maximal ideal, and let $B=A/mathfrak{m}$. Then, the canonical morphism $Arightarrow B$ is flat but not injective.
answered Nov 22 at 18:13
Fred Rohrer
261110
add a comment |
up vote
0
down vote
Let $A$ be a nonzero and absolutely flat ring, let $mathfrak{m}subseteq A$ be a maximal ideal, and let $B=A/mathfrak{m}$. Then, the canonical morphism $Arightarrow B$ is flat but not injective.
answered Nov 22 at 18:13
Fred Rohrer
261110
add a comment |
up vote
0
down vote
up vote
0
down vote
Let $A$ be a nonzero and absolutely flat ring, let $mathfrak{m}subseteq A$ be a maximal ideal, and let $B=A/mathfrak{m}$. Then, the canonical morphism $Arightarrow B$ is flat but not injective.
answered Nov 22 at 18:13
Fred Rohrer
261110
Let $A$ be a nonzero and absolutely flat ring, let $mathfrak{m}subseteq A$ be a maximal ideal, and let $B=A/mathfrak{m}$. Then, the canonical morphism $Arightarrow B$ is flat but not injective.
answered Nov 22 at 18:13
Fred Rohrer
261110
answered Nov 22 at 18:13
Fred Rohrer
261110
answered Nov 22 at 18:13
Fred Rohrer
261110
answered Nov 22 at 18:13
Fred Rohrer
261110
261110
add a comment |
add a comment |
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1
$A$ any ring, $B$ the zero ring?
– Lord Shark the Unknown
Oct 4 at 18:01
@LordSharktheUnknown $A,B$ are unital rings.
– user45765
Oct 4 at 18:02
4
The zero ring is unital....
– Lord Shark the Unknown
Oct 4 at 18:03
@LordSharktheUnknown Can I have a unital ring counter example rather than $0$ ring?
– user45765
Oct 4 at 18:05
3
Take $A=Boplus B$ and $Ato B$ the projection onto the first factor.
– David Hill
Oct 4 at 18:05