Example of a one dimensional GCD domain which is not a UFD.
0
$begingroup$
I know that every UFD is a GCD domain. But every GCD domain is not a UFD.
I want to make sure that a one dimensional GCD domain is not necessarily a UFD, so I'm looking for an example to confirm that.
Please help me.
commutative-algebra examples-counterexamples greatest-common-divisor unique-factorization-domains
edited Dec 15 '18 at 8:02
user26857
39.3k124183
asked Dec 14 '18 at 11:09
REZAREZA
91
$endgroup$
add a comment |
0
$begingroup$
I know that every UFD is a GCD domain. But every GCD domain is not a UFD.
I want to make sure that a one dimensional GCD domain is not necessarily a UFD, so I'm looking for an example to confirm that.
Please help me.
commutative-algebra examples-counterexamples greatest-common-divisor unique-factorization-domains
edited Dec 15 '18 at 8:02
user26857
39.3k124183
asked Dec 14 '18 at 11:09
REZAREZA
91
$endgroup$
$begingroup$
Very related post.
$endgroup$
– Arthur
Dec 14 '18 at 11:17
$begingroup$
Possible duplicate of Looking for an example of a GCD domain which is not a UFD
$endgroup$
– Tony S.F.
Dec 14 '18 at 12:19
1
$begingroup$
In dimension $1!: $ UFD $iff$ PID and GCD $iff$ Bezout, so you seek a Bezout domain that is not a PID, i.e. one where ACCP fails.
$endgroup$
– Bill Dubuque
Dec 14 '18 at 15:34
$begingroup$
Some hints for proving that a $1$-dimensional GCD domain is Bezout: locally it is a valuation domain (see here), hence Prufer, but a Prufer GCD domain is Bezout (see here).
$endgroup$
– user26857
Dec 15 '18 at 8:06
add a comment |
0
0
0
$begingroup$
I know that every UFD is a GCD domain. But every GCD domain is not a UFD.
I want to make sure that a one dimensional GCD domain is not necessarily a UFD, so I'm looking for an example to confirm that.
Please help me.
commutative-algebra examples-counterexamples greatest-common-divisor unique-factorization-domains
edited Dec 15 '18 at 8:02
user26857
39.3k124183
asked Dec 14 '18 at 11:09
REZAREZA
91
$endgroup$
I know that every UFD is a GCD domain. But every GCD domain is not a UFD.
I want to make sure that a one dimensional GCD domain is not necessarily a UFD, so I'm looking for an example to confirm that.
Please help me.
commutative-algebra examples-counterexamples greatest-common-divisor unique-factorization-domains
commutative-algebra examples-counterexamples greatest-common-divisor unique-factorization-domains
edited Dec 15 '18 at 8:02
user26857
39.3k124183
asked Dec 14 '18 at 11:09
REZAREZA
91
edited Dec 15 '18 at 8:02
user26857
39.3k124183
asked Dec 14 '18 at 11:09
REZAREZA
91
edited Dec 15 '18 at 8:02
user26857
39.3k124183
edited Dec 15 '18 at 8:02
user26857
39.3k124183
edited Dec 15 '18 at 8:02
user26857
39.3k124183
39.3k124183
asked Dec 14 '18 at 11:09
REZAREZA
91
asked Dec 14 '18 at 11:09
REZAREZA
91
asked Dec 14 '18 at 11:09
REZAREZA
91
91
$begingroup$
Very related post.
$endgroup$
– Arthur
Dec 14 '18 at 11:17
$begingroup$
Possible duplicate of Looking for an example of a GCD domain which is not a UFD
$endgroup$
– Tony S.F.
Dec 14 '18 at 12:19
1
$begingroup$
In dimension $1!: $ UFD $iff$ PID and GCD $iff$ Bezout, so you seek a Bezout domain that is not a PID, i.e. one where ACCP fails.
$endgroup$
– Bill Dubuque
Dec 14 '18 at 15:34
$begingroup$
Some hints for proving that a $1$-dimensional GCD domain is Bezout: locally it is a valuation domain (see here), hence Prufer, but a Prufer GCD domain is Bezout (see here).
$endgroup$
– user26857
Dec 15 '18 at 8:06
add a comment |
$begingroup$
Very related post.
$endgroup$
– Arthur
Dec 14 '18 at 11:17
$begingroup$
Possible duplicate of Looking for an example of a GCD domain which is not a UFD
$endgroup$
– Tony S.F.
Dec 14 '18 at 12:19
1
$begingroup$
In dimension $1!: $ UFD $iff$ PID and GCD $iff$ Bezout, so you seek a Bezout domain that is not a PID, i.e. one where ACCP fails.
$endgroup$
– Bill Dubuque
Dec 14 '18 at 15:34
$begingroup$
Some hints for proving that a $1$-dimensional GCD domain is Bezout: locally it is a valuation domain (see here), hence Prufer, but a Prufer GCD domain is Bezout (see here).
$endgroup$
– user26857
Dec 15 '18 at 8:06
$begingroup$
Very related post.
$endgroup$
– Arthur
Dec 14 '18 at 11:17
$begingroup$
Very related post.
$endgroup$
– Arthur
Dec 14 '18 at 11:17
$begingroup$
Possible duplicate of Looking for an example of a GCD domain which is not a UFD
$endgroup$
– Tony S.F.
Dec 14 '18 at 12:19
$begingroup$
Possible duplicate of Looking for an example of a GCD domain which is not a UFD
$endgroup$
– Tony S.F.
Dec 14 '18 at 12:19
1
1
$begingroup$
In dimension $1!: $ UFD $iff$ PID and GCD $iff$ Bezout, so you seek a Bezout domain that is not a PID, i.e. one where ACCP fails.
$endgroup$
– Bill Dubuque
Dec 14 '18 at 15:34
$begingroup$
In dimension $1!: $ UFD $iff$ PID and GCD $iff$ Bezout, so you seek a Bezout domain that is not a PID, i.e. one where ACCP fails.
$endgroup$
– Bill Dubuque
Dec 14 '18 at 15:34
$begingroup$
Some hints for proving that a $1$-dimensional GCD domain is Bezout: locally it is a valuation domain (see here), hence Prufer, but a Prufer GCD domain is Bezout (see here).
$endgroup$
– user26857
Dec 15 '18 at 8:06
$begingroup$
Some hints for proving that a $1$-dimensional GCD domain is Bezout: locally it is a valuation domain (see here), hence Prufer, but a Prufer GCD domain is Bezout (see here).
$endgroup$
– user26857
Dec 15 '18 at 8:06
add a comment |
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$begingroup$
Very related post.
$endgroup$
– Arthur
Dec 14 '18 at 11:17
$begingroup$
Possible duplicate of Looking for an example of a GCD domain which is not a UFD
$endgroup$
– Tony S.F.
Dec 14 '18 at 12:19
1
$begingroup$
In dimension $1!: $ UFD $iff$ PID and GCD $iff$ Bezout, so you seek a Bezout domain that is not a PID, i.e. one where ACCP fails.
$endgroup$
– Bill Dubuque
Dec 14 '18 at 15:34
$begingroup$
Some hints for proving that a $1$-dimensional GCD domain is Bezout: locally it is a valuation domain (see here), hence Prufer, but a Prufer GCD domain is Bezout (see here).
$endgroup$
– user26857
Dec 15 '18 at 8:06