Infinite Noetherian ring of dimension $1$ in which distinct non-zero ideals have distinct and finite index
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Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an integral domain. Now also assume that distinct ideals have distinct index; then is it true that $R$ is a UFD, or at least normal (integrally closed in its fraction field) ?
(note: $dim R le 1$ , now if $dim R=0$, then $R$ is an Artinian domain, so a field. So w.l.o.g., assume $R$ has dimension $1$ )
algebraic-geometry ring-theory commutative-algebra dimension-theory integral-extensions
asked Dec 31 '18 at 1:25
user521337user521337
1,2061417
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add a comment |
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Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an integral domain. Now also assume that distinct ideals have distinct index; then is it true that $R$ is a UFD, or at least normal (integrally closed in its fraction field) ?
(note: $dim R le 1$ , now if $dim R=0$, then $R$ is an Artinian domain, so a field. So w.l.o.g., assume $R$ has dimension $1$ )
algebraic-geometry ring-theory commutative-algebra dimension-theory integral-extensions
asked Dec 31 '18 at 1:25
user521337user521337
1,2061417
$endgroup$
$begingroup$
Cross-posted to MO here: mathoverflow.net/questions/320068/…
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– Alex Wertheim
Jan 7 at 1:58
add a comment |
$begingroup$
Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an integral domain. Now also assume that distinct ideals have distinct index; then is it true that $R$ is a UFD, or at least normal (integrally closed in its fraction field) ?
(note: $dim R le 1$ , now if $dim R=0$, then $R$ is an Artinian domain, so a field. So w.l.o.g., assume $R$ has dimension $1$ )
algebraic-geometry ring-theory commutative-algebra dimension-theory integral-extensions
asked Dec 31 '18 at 1:25
user521337user521337
1,2061417
$endgroup$
Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an integral domain. Now also assume that distinct ideals have distinct index; then is it true that $R$ is a UFD, or at least normal (integrally closed in its fraction field) ?
(note: $dim R le 1$ , now if $dim R=0$, then $R$ is an Artinian domain, so a field. So w.l.o.g., assume $R$ has dimension $1$ )
algebraic-geometry ring-theory commutative-algebra dimension-theory integral-extensions
algebraic-geometry ring-theory commutative-algebra dimension-theory integral-extensions
asked Dec 31 '18 at 1:25
user521337user521337
1,2061417
asked Dec 31 '18 at 1:25
user521337user521337
1,2061417
edited Dec 31 '18 at 5:59
user521337
asked Dec 31 '18 at 1:25
user521337user521337
1,2061417
asked Dec 31 '18 at 1:25
user521337user521337
1,2061417
asked Dec 31 '18 at 1:25
user521337user521337
1,2061417
1,2061417
$begingroup$
Cross-posted to MO here: mathoverflow.net/questions/320068/…
$endgroup$
– Alex Wertheim
Jan 7 at 1:58
add a comment |
$begingroup$
Cross-posted to MO here: mathoverflow.net/questions/320068/…
$endgroup$
– Alex Wertheim
Jan 7 at 1:58
$begingroup$
Cross-posted to MO here: mathoverflow.net/questions/320068/…
$endgroup$
– Alex Wertheim
Jan 7 at 1:58
$begingroup$
Cross-posted to MO here: mathoverflow.net/questions/320068/…
$endgroup$
– Alex Wertheim
Jan 7 at 1:58
add a comment |
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$begingroup$
Cross-posted to MO here: mathoverflow.net/questions/320068/…
$endgroup$
– Alex Wertheim
Jan 7 at 1:58