Mixing
$Z[(I, J)]$ is the set of all $Z_1 cap Z_2 $, where $Z_1 in Z[I]$ and $Z_2 in Z[J]$
$begingroup$
Definition 1:
$Z(f)= { x in X : f(x) = 0 }$ is a zero-set.($f in C(X)$)
$Z(X) ={ Z(f): f in C (X)}$ ($C(X)$ is the ring of real continuous function on $X$.)
Definition 2:
A nonempty subfamily $mathbf{F} in Z(X)$ is called a $Z$-filter provided that:
1: $emptyset notin mathbf{F}$
2:if $z_1, z_2 in mathbf{F}$, then $z_1 cap z_2 in mathbf{F}$
3:if $z in mathbf{F}, z^{*} in Z(X)$, and $z^{*} supset Z$, then $z^{*} in mathbf{F}$
Definition 3:
$ Z[I] = { Z(f) : f in I } quad ( I in C(X) )$ is a $Z$-filter and $I$ is an ideal.
$(I, J)$ is Ideal generated by $ I, J$.
So, my question:
Why is $Z[(I, J)]$ the set of all $Z_1 cap Z_2 $, where $Z_1 in Z[I]$ and $Z_2 in Z[J]$?
ring-theory ideals topological-groups filters
$endgroup$
add a comment |
$begingroup$
Definition 1:
$Z(f)= { x in X : f(x) = 0 }$ is a zero-set.($f in C(X)$)
$Z(X) ={ Z(f): f in C (X)}$ ($C(X)$ is the ring of real continuous function on $X$.)
Definition 2:
A nonempty subfamily $mathbf{F} in Z(X)$ is called a $Z$-filter provided that:
1: $emptyset notin mathbf{F}$
2:if $z_1, z_2 in mathbf{F}$, then $z_1 cap z_2 in mathbf{F}$
3:if $z in mathbf{F}, z^{*} in Z(X)$, and $z^{*} supset Z$, then $z^{*} in mathbf{F}$
Definition 3:
$ Z[I] = { Z(f) : f in I } quad ( I in C(X) )$ is a $Z$-filter and $I$ is an ideal.
$(I, J)$ is Ideal generated by $ I, J$.
So, my question:
Why is $Z[(I, J)]$ the set of all $Z_1 cap Z_2 $, where $Z_1 in Z[I]$ and $Z_2 in Z[J]$?
ring-theory ideals topological-groups filters
$endgroup$
$begingroup$
What is $Z$? What is $X$? What is $Z(X)$? In othe words, what kind of objects are $z_1,z_2$? Is $phi$ in 1. meant to be the empty set,emptyset$emptyset$ orvarnothing$varnothing$, rather than a distinguished element named after the Greek letter? What is $C(x)$? You are assuming a lot of notation that is not evident.
$endgroup$
– Arturo Magidin
Jan 10 at 19:29
$begingroup$
Note that in general, $z$ and $Z$, $x$ and $X$, etc., are considered different things. So in your item (3), you have a $z$ that is never mentioned again, and you have a $Z$ that appears from nowhere... I'm guessing those are supposed to be the same, but you write them as different items.
$endgroup$
– Arturo Magidin
Jan 10 at 19:55
add a comment |
$begingroup$
Definition 1:
$Z(f)= { x in X : f(x) = 0 }$ is a zero-set.($f in C(X)$)
$Z(X) ={ Z(f): f in C (X)}$ ($C(X)$ is the ring of real continuous function on $X$.)
Definition 2:
A nonempty subfamily $mathbf{F} in Z(X)$ is called a $Z$-filter provided that:
1: $emptyset notin mathbf{F}$
2:if $z_1, z_2 in mathbf{F}$, then $z_1 cap z_2 in mathbf{F}$
3:if $z in mathbf{F}, z^{*} in Z(X)$, and $z^{*} supset Z$, then $z^{*} in mathbf{F}$
Definition 3:
$ Z[I] = { Z(f) : f in I } quad ( I in C(X) )$ is a $Z$-filter and $I$ is an ideal.
$(I, J)$ is Ideal generated by $ I, J$.
So, my question:
Why is $Z[(I, J)]$ the set of all $Z_1 cap Z_2 $, where $Z_1 in Z[I]$ and $Z_2 in Z[J]$?
ring-theory ideals topological-groups filters
$endgroup$
Definition 1:
$Z(f)= { x in X : f(x) = 0 }$ is a zero-set.($f in C(X)$)
$Z(X) ={ Z(f): f in C (X)}$ ($C(X)$ is the ring of real continuous function on $X$.)
Definition 2:
A nonempty subfamily $mathbf{F} in Z(X)$ is called a $Z$-filter provided that:
1: $emptyset notin mathbf{F}$
2:if $z_1, z_2 in mathbf{F}$, then $z_1 cap z_2 in mathbf{F}$
3:if $z in mathbf{F}, z^{*} in Z(X)$, and $z^{*} supset Z$, then $z^{*} in mathbf{F}$
Definition 3:
$ Z[I] = { Z(f) : f in I } quad ( I in C(X) )$ is a $Z$-filter and $I$ is an ideal.
$(I, J)$ is Ideal generated by $ I, J$.
So, my question:
Why is $Z[(I, J)]$ the set of all $Z_1 cap Z_2 $, where $Z_1 in Z[I]$ and $Z_2 in Z[J]$?
ring-theory ideals topological-groups filters
ring-theory ideals topological-groups filters
edited Jan 10 at 20:16
asked Jan 10 at 18:18
user633199
$begingroup$
What is $Z$? What is $X$? What is $Z(X)$? In othe words, what kind of objects are $z_1,z_2$? Is $phi$ in 1. meant to be the empty set,emptyset$emptyset$ orvarnothing$varnothing$, rather than a distinguished element named after the Greek letter? What is $C(x)$? You are assuming a lot of notation that is not evident.
$endgroup$
– Arturo Magidin
Jan 10 at 19:29
$begingroup$
Note that in general, $z$ and $Z$, $x$ and $X$, etc., are considered different things. So in your item (3), you have a $z$ that is never mentioned again, and you have a $Z$ that appears from nowhere... I'm guessing those are supposed to be the same, but you write them as different items.
$endgroup$
– Arturo Magidin
Jan 10 at 19:55
add a comment |
$begingroup$
What is $Z$? What is $X$? What is $Z(X)$? In othe words, what kind of objects are $z_1,z_2$? Is $phi$ in 1. meant to be the empty set,emptyset$emptyset$ orvarnothing$varnothing$, rather than a distinguished element named after the Greek letter? What is $C(x)$? You are assuming a lot of notation that is not evident.
$endgroup$
– Arturo Magidin
Jan 10 at 19:29
$begingroup$
Note that in general, $z$ and $Z$, $x$ and $X$, etc., are considered different things. So in your item (3), you have a $z$ that is never mentioned again, and you have a $Z$ that appears from nowhere... I'm guessing those are supposed to be the same, but you write them as different items.
$endgroup$
– Arturo Magidin
Jan 10 at 19:55
$begingroup$
What is $Z$? What is $X$? What is $Z(X)$? In othe words, what kind of objects are $z_1,z_2$? Is $phi$ in 1. meant to be the empty set,
emptyset $emptyset$ or varnothing $varnothing$, rather than a distinguished element named after the Greek letter? What is $C(x)$? You are assuming a lot of notation that is not evident.$endgroup$
– Arturo Magidin
Jan 10 at 19:29
$begingroup$
What is $Z$? What is $X$? What is $Z(X)$? In othe words, what kind of objects are $z_1,z_2$? Is $phi$ in 1. meant to be the empty set,
emptyset $emptyset$ or varnothing $varnothing$, rather than a distinguished element named after the Greek letter? What is $C(x)$? You are assuming a lot of notation that is not evident.$endgroup$
– Arturo Magidin
Jan 10 at 19:29
$begingroup$
Note that in general, $z$ and $Z$, $x$ and $X$, etc., are considered different things. So in your item (3), you have a $z$ that is never mentioned again, and you have a $Z$ that appears from nowhere... I'm guessing those are supposed to be the same, but you write them as different items.
$endgroup$
– Arturo Magidin
Jan 10 at 19:55
$begingroup$
Note that in general, $z$ and $Z$, $x$ and $X$, etc., are considered different things. So in your item (3), you have a $z$ that is never mentioned again, and you have a $Z$ that appears from nowhere... I'm guessing those are supposed to be the same, but you write them as different items.
$endgroup$
– Arturo Magidin
Jan 10 at 19:55
add a comment |
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$begingroup$
What is $Z$? What is $X$? What is $Z(X)$? In othe words, what kind of objects are $z_1,z_2$? Is $phi$ in 1. meant to be the empty set,
emptyset$emptyset$ orvarnothing$varnothing$, rather than a distinguished element named after the Greek letter? What is $C(x)$? You are assuming a lot of notation that is not evident.$endgroup$
– Arturo Magidin
Jan 10 at 19:29
$begingroup$
Note that in general, $z$ and $Z$, $x$ and $X$, etc., are considered different things. So in your item (3), you have a $z$ that is never mentioned again, and you have a $Z$ that appears from nowhere... I'm guessing those are supposed to be the same, but you write them as different items.
$endgroup$
– Arturo Magidin
Jan 10 at 19:55