Maximal ideal with respect to property that it does not contain an ideal A
up vote
1
down vote
favorite
let A=($a1, a2, .... ,an$) be a non-zero finitely generated ideal of R.
Prove that there is an ideal B which is maximal with respect to property that it does not contain A.
I don't know how to proceed. Maybe by Zorn's lemma.
If I define order by containment of ideals. Then in a chain what will be the maximal element.
If ∪J { J ∈ Chain } how to check that A is not contained inside ∪J.
ring-theory maximal-and-prime-ideals
add a comment |
up vote
1
down vote
favorite
let A=($a1, a2, .... ,an$) be a non-zero finitely generated ideal of R.
Prove that there is an ideal B which is maximal with respect to property that it does not contain A.
I don't know how to proceed. Maybe by Zorn's lemma.
If I define order by containment of ideals. Then in a chain what will be the maximal element.
If ∪J { J ∈ Chain } how to check that A is not contained inside ∪J.
ring-theory maximal-and-prime-ideals
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
let A=($a1, a2, .... ,an$) be a non-zero finitely generated ideal of R.
Prove that there is an ideal B which is maximal with respect to property that it does not contain A.
I don't know how to proceed. Maybe by Zorn's lemma.
If I define order by containment of ideals. Then in a chain what will be the maximal element.
If ∪J { J ∈ Chain } how to check that A is not contained inside ∪J.
ring-theory maximal-and-prime-ideals
let A=($a1, a2, .... ,an$) be a non-zero finitely generated ideal of R.
Prove that there is an ideal B which is maximal with respect to property that it does not contain A.
I don't know how to proceed. Maybe by Zorn's lemma.
If I define order by containment of ideals. Then in a chain what will be the maximal element.
If ∪J { J ∈ Chain } how to check that A is not contained inside ∪J.
ring-theory maximal-and-prime-ideals
ring-theory maximal-and-prime-ideals
edited Nov 16 at 5:38
asked Nov 16 at 4:39
infintedimensional
428
428
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
You are right, the idea, is the Zorn, consider the set $J$ of ideals which does not contain $A$, let $(B_i)$ be a family of well ordered (by the inclusion) element of $J$, $B=cup_{iin I}B_i$ is the sup. Suppose that $B$ contains $A$, $a_jin B_{i_j}$ let $k=sup(i_1,...,i_n)$, since $B_{i_j}subset B_k$, $a_1,...,a_nin B_k$ contradiction. Thus $J$ has a maximum element.
can you please explain last line saying k = sup(i1, ....in )
– infintedimensional
Nov 16 at 4:49
since $J$ is totally ordered, $k$ exists.
– Tsemo Aristide
Nov 16 at 4:50
got it. Thank you !
– infintedimensional
Nov 16 at 4:50
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
You are right, the idea, is the Zorn, consider the set $J$ of ideals which does not contain $A$, let $(B_i)$ be a family of well ordered (by the inclusion) element of $J$, $B=cup_{iin I}B_i$ is the sup. Suppose that $B$ contains $A$, $a_jin B_{i_j}$ let $k=sup(i_1,...,i_n)$, since $B_{i_j}subset B_k$, $a_1,...,a_nin B_k$ contradiction. Thus $J$ has a maximum element.
can you please explain last line saying k = sup(i1, ....in )
– infintedimensional
Nov 16 at 4:49
since $J$ is totally ordered, $k$ exists.
– Tsemo Aristide
Nov 16 at 4:50
got it. Thank you !
– infintedimensional
Nov 16 at 4:50
add a comment |
up vote
1
down vote
You are right, the idea, is the Zorn, consider the set $J$ of ideals which does not contain $A$, let $(B_i)$ be a family of well ordered (by the inclusion) element of $J$, $B=cup_{iin I}B_i$ is the sup. Suppose that $B$ contains $A$, $a_jin B_{i_j}$ let $k=sup(i_1,...,i_n)$, since $B_{i_j}subset B_k$, $a_1,...,a_nin B_k$ contradiction. Thus $J$ has a maximum element.
can you please explain last line saying k = sup(i1, ....in )
– infintedimensional
Nov 16 at 4:49
since $J$ is totally ordered, $k$ exists.
– Tsemo Aristide
Nov 16 at 4:50
got it. Thank you !
– infintedimensional
Nov 16 at 4:50
add a comment |
up vote
1
down vote
up vote
1
down vote
You are right, the idea, is the Zorn, consider the set $J$ of ideals which does not contain $A$, let $(B_i)$ be a family of well ordered (by the inclusion) element of $J$, $B=cup_{iin I}B_i$ is the sup. Suppose that $B$ contains $A$, $a_jin B_{i_j}$ let $k=sup(i_1,...,i_n)$, since $B_{i_j}subset B_k$, $a_1,...,a_nin B_k$ contradiction. Thus $J$ has a maximum element.
You are right, the idea, is the Zorn, consider the set $J$ of ideals which does not contain $A$, let $(B_i)$ be a family of well ordered (by the inclusion) element of $J$, $B=cup_{iin I}B_i$ is the sup. Suppose that $B$ contains $A$, $a_jin B_{i_j}$ let $k=sup(i_1,...,i_n)$, since $B_{i_j}subset B_k$, $a_1,...,a_nin B_k$ contradiction. Thus $J$ has a maximum element.
answered Nov 16 at 4:45
Tsemo Aristide
54.4k11344
54.4k11344
can you please explain last line saying k = sup(i1, ....in )
– infintedimensional
Nov 16 at 4:49
since $J$ is totally ordered, $k$ exists.
– Tsemo Aristide
Nov 16 at 4:50
got it. Thank you !
– infintedimensional
Nov 16 at 4:50
add a comment |
can you please explain last line saying k = sup(i1, ....in )
– infintedimensional
Nov 16 at 4:49
since $J$ is totally ordered, $k$ exists.
– Tsemo Aristide
Nov 16 at 4:50
got it. Thank you !
– infintedimensional
Nov 16 at 4:50
can you please explain last line saying k = sup(i1, ....in )
– infintedimensional
Nov 16 at 4:49
can you please explain last line saying k = sup(i1, ....in )
– infintedimensional
Nov 16 at 4:49
since $J$ is totally ordered, $k$ exists.
– Tsemo Aristide
Nov 16 at 4:50
since $J$ is totally ordered, $k$ exists.
– Tsemo Aristide
Nov 16 at 4:50
got it. Thank you !
– infintedimensional
Nov 16 at 4:50
got it. Thank you !
– infintedimensional
Nov 16 at 4:50
add a comment |
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3000724%2fmaximal-ideal-with-respect-to-property-that-it-does-not-contain-an-ideal-a%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown