induced maximal ideals in ring of functions [closed]
$begingroup$
Let $R$ be a ring and $X$ be a non-empty set.
We know that $operatorname{Map}(X,R) = {f : X to R}$ is a ring
where, $(f+g)(x)= f(x) + g(x)$ and $(fcdot g)(x)=f(x)cdot g(x)$
I want to know that if $M$ is a maximal ideal in $R$ whether this ideal induces a maximal ideal in $operatorname{Map}(X,R)$?
abstract-algebra
edited Dec 26 '18 at 21:05
user26857
39.3k124183
asked Dec 26 '18 at 14:27
S. WalkerS. Walker
92
$endgroup$
closed as off-topic by amWhy, user26857, Saad, KReiser, mrtaurho Dec 27 '18 at 12:37
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – amWhy, user26857, Saad, KReiser, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Let $R$ be a ring and $X$ be a non-empty set.
We know that $operatorname{Map}(X,R) = {f : X to R}$ is a ring
where, $(f+g)(x)= f(x) + g(x)$ and $(fcdot g)(x)=f(x)cdot g(x)$
I want to know that if $M$ is a maximal ideal in $R$ whether this ideal induces a maximal ideal in $operatorname{Map}(X,R)$?
abstract-algebra
edited Dec 26 '18 at 21:05
user26857
39.3k124183
asked Dec 26 '18 at 14:27
S. WalkerS. Walker
92
$endgroup$
closed as off-topic by amWhy, user26857, Saad, KReiser, mrtaurho Dec 27 '18 at 12:37
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – amWhy, user26857, Saad, KReiser, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Let $R$ be a ring and $X$ be a non-empty set.
We know that $operatorname{Map}(X,R) = {f : X to R}$ is a ring
where, $(f+g)(x)= f(x) + g(x)$ and $(fcdot g)(x)=f(x)cdot g(x)$
I want to know that if $M$ is a maximal ideal in $R$ whether this ideal induces a maximal ideal in $operatorname{Map}(X,R)$?
abstract-algebra
edited Dec 26 '18 at 21:05
user26857
39.3k124183
asked Dec 26 '18 at 14:27
S. WalkerS. Walker
92
$endgroup$
Let $R$ be a ring and $X$ be a non-empty set.
We know that $operatorname{Map}(X,R) = {f : X to R}$ is a ring
where, $(f+g)(x)= f(x) + g(x)$ and $(fcdot g)(x)=f(x)cdot g(x)$
I want to know that if $M$ is a maximal ideal in $R$ whether this ideal induces a maximal ideal in $operatorname{Map}(X,R)$?
abstract-algebra
abstract-algebra
edited Dec 26 '18 at 21:05
user26857
39.3k124183
asked Dec 26 '18 at 14:27
S. WalkerS. Walker
92
edited Dec 26 '18 at 21:05
user26857
39.3k124183
asked Dec 26 '18 at 14:27
S. WalkerS. Walker
92
edited Dec 26 '18 at 21:05
user26857
39.3k124183
edited Dec 26 '18 at 21:05
user26857
39.3k124183
edited Dec 26 '18 at 21:05
user26857
39.3k124183
39.3k124183
asked Dec 26 '18 at 14:27
S. WalkerS. Walker
92
asked Dec 26 '18 at 14:27
S. WalkerS. Walker
92
asked Dec 26 '18 at 14:27
S. WalkerS. Walker
92
92
closed as off-topic by amWhy, user26857, Saad, KReiser, mrtaurho Dec 27 '18 at 12:37
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – amWhy, user26857, Saad, KReiser, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by amWhy, user26857, Saad, KReiser, mrtaurho Dec 27 '18 at 12:37
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – amWhy, user26857, Saad, KReiser, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Not only one, but sometimes many. For each $xin X$ and maximal ideal $M$, you can consider the subset of functions $f$ such that $f(x)in M$. If you check, you'll find this is a maximal ideal of the set of functions.
It is not always the case that these are all the maximal ideals, though.
edited Dec 26 '18 at 21:05
user26857
39.3k124183
answered Dec 26 '18 at 15:14
rschwiebrschwieb
107k12102251
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Not only one, but sometimes many. For each $xin X$ and maximal ideal $M$, you can consider the subset of functions $f$ such that $f(x)in M$. If you check, you'll find this is a maximal ideal of the set of functions.
It is not always the case that these are all the maximal ideals, though.
edited Dec 26 '18 at 21:05
user26857
39.3k124183
answered Dec 26 '18 at 15:14
rschwiebrschwieb
107k12102251
$endgroup$
add a comment |
$begingroup$
Not only one, but sometimes many. For each $xin X$ and maximal ideal $M$, you can consider the subset of functions $f$ such that $f(x)in M$. If you check, you'll find this is a maximal ideal of the set of functions.
It is not always the case that these are all the maximal ideals, though.
edited Dec 26 '18 at 21:05
user26857
39.3k124183
answered Dec 26 '18 at 15:14
rschwiebrschwieb
107k12102251
$endgroup$
add a comment |
$begingroup$
Not only one, but sometimes many. For each $xin X$ and maximal ideal $M$, you can consider the subset of functions $f$ such that $f(x)in M$. If you check, you'll find this is a maximal ideal of the set of functions.
It is not always the case that these are all the maximal ideals, though.
edited Dec 26 '18 at 21:05
user26857
39.3k124183
answered Dec 26 '18 at 15:14
rschwiebrschwieb
107k12102251
$endgroup$
Not only one, but sometimes many. For each $xin X$ and maximal ideal $M$, you can consider the subset of functions $f$ such that $f(x)in M$. If you check, you'll find this is a maximal ideal of the set of functions.
It is not always the case that these are all the maximal ideals, though.
edited Dec 26 '18 at 21:05
user26857
39.3k124183
answered Dec 26 '18 at 15:14
rschwiebrschwieb
107k12102251
edited Dec 26 '18 at 21:05
user26857
39.3k124183
edited Dec 26 '18 at 21:05
user26857
39.3k124183
edited Dec 26 '18 at 21:05
user26857
39.3k124183
39.3k124183
answered Dec 26 '18 at 15:14
rschwiebrschwieb
107k12102251
answered Dec 26 '18 at 15:14
rschwiebrschwieb
107k12102251
answered Dec 26 '18 at 15:14
rschwiebrschwieb
107k12102251
107k12102251
add a comment |
add a comment |
