Problem understanding the concept of principal ideal
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Definition of Principal ideal:
Let $R$ be a commutative ring with unity and let $a in R$ . The set $langle arangle = ramid r in R$ is an ideal of $R$ called the principal ideal generated by a.
Doubt
What is the principal ideal in case of a subset of $R[x]$ (set of all polynomials with real coefficients) with constant term zero.
$A=langle xrangle $ will work but I actually don't understand what $langle xrangle$ mean here.
I think
$$A=f(x) in R[x]mid f(0)=0.$$
How all these things fit together?
abstract-algebra ring-theory ideals
edited Sep 6 at 9:22
Arnaud D.
14.9k52142
asked Sep 6 at 9:13
blue boy
1,117513
 |Â
show 2 more comments
up vote
0
down vote
favorite
Definition of Principal ideal:
Let $R$ be a commutative ring with unity and let $a in R$ . The set $langle arangle = ramid r in R$ is an ideal of $R$ called the principal ideal generated by a.
Doubt
What is the principal ideal in case of a subset of $R[x]$ (set of all polynomials with real coefficients) with constant term zero.
$A=langle xrangle $ will work but I actually don't understand what $langle xrangle$ mean here.
I think
$$A=f(x) in R[x]mid f(0)=0.$$
How all these things fit together?
abstract-algebra ring-theory ideals
edited Sep 6 at 9:22
Arnaud D.
14.9k52142
asked Sep 6 at 9:13
blue boy
1,117513
2
You wrote yourself what $langle xrangle $ means just above it.
– Tobias Kildetoft
Sep 6 at 9:17
Ok. What if $angle x^2+1angle$ means then ?
– blue boy
Sep 6 at 9:19
2
Still, you have the definition right there, so I am not sure what you are really asking about.
– Tobias Kildetoft
Sep 6 at 9:21
@TobiasKildetoft $<x^2+1>=f(x) in R[x]$.
– blue boy
Sep 6 at 9:26
2
You would make things more clear for yourself if you stop writing $f(x)$ for the polynomial $f$, since that makes it hard to tell the difference between evaluating $f$ at $x^2+1$ and multiplying $f$ by $x^2+1$.
– Tobias Kildetoft
Sep 6 at 9:31
 |Â
show 2 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Definition of Principal ideal:
Let $R$ be a commutative ring with unity and let $a in R$ . The set $langle arangle = ramid r in R$ is an ideal of $R$ called the principal ideal generated by a.
Doubt
What is the principal ideal in case of a subset of $R[x]$ (set of all polynomials with real coefficients) with constant term zero.
$A=langle xrangle $ will work but I actually don't understand what $langle xrangle$ mean here.
I think
$$A=f(x) in R[x]mid f(0)=0.$$
How all these things fit together?
abstract-algebra ring-theory ideals
edited Sep 6 at 9:22
Arnaud D.
14.9k52142
asked Sep 6 at 9:13
blue boy
1,117513
Definition of Principal ideal:
Let $R$ be a commutative ring with unity and let $a in R$ . The set $langle arangle = ramid r in R$ is an ideal of $R$ called the principal ideal generated by a.
Doubt
What is the principal ideal in case of a subset of $R[x]$ (set of all polynomials with real coefficients) with constant term zero.
$A=langle xrangle $ will work but I actually don't understand what $langle xrangle$ mean here.
I think
$$A=f(x) in R[x]mid f(0)=0.$$
How all these things fit together?
abstract-algebra ring-theory ideals
abstract-algebra ring-theory ideals
edited Sep 6 at 9:22
Arnaud D.
14.9k52142
asked Sep 6 at 9:13
blue boy
1,117513
edited Sep 6 at 9:22
Arnaud D.
14.9k52142
asked Sep 6 at 9:13
blue boy
1,117513
edited Sep 6 at 9:22
Arnaud D.
14.9k52142
edited Sep 6 at 9:22
Arnaud D.
14.9k52142
edited Sep 6 at 9:22
Arnaud D.
14.9k52142
14.9k52142
asked Sep 6 at 9:13
blue boy
1,117513
asked Sep 6 at 9:13
blue boy
1,117513
asked Sep 6 at 9:13
blue boy
1,117513
1,117513
2
You wrote yourself what $langle xrangle $ means just above it.
– Tobias Kildetoft
Sep 6 at 9:17
Ok. What if $angle x^2+1angle$ means then ?
– blue boy
Sep 6 at 9:19
2
Still, you have the definition right there, so I am not sure what you are really asking about.
– Tobias Kildetoft
Sep 6 at 9:21
@TobiasKildetoft $<x^2+1>=f(x) in R[x]$.
– blue boy
Sep 6 at 9:26
2
You would make things more clear for yourself if you stop writing $f(x)$ for the polynomial $f$, since that makes it hard to tell the difference between evaluating $f$ at $x^2+1$ and multiplying $f$ by $x^2+1$.
– Tobias Kildetoft
Sep 6 at 9:31
 |Â
show 2 more comments
2
You wrote yourself what $langle xrangle $ means just above it.
– Tobias Kildetoft
Sep 6 at 9:17
Ok. What if $angle x^2+1angle$ means then ?
– blue boy
Sep 6 at 9:19
2
Still, you have the definition right there, so I am not sure what you are really asking about.
– Tobias Kildetoft
Sep 6 at 9:21
@TobiasKildetoft $<x^2+1>=f(x) in R[x]$.
– blue boy
Sep 6 at 9:26
2
You would make things more clear for yourself if you stop writing $f(x)$ for the polynomial $f$, since that makes it hard to tell the difference between evaluating $f$ at $x^2+1$ and multiplying $f$ by $x^2+1$.
– Tobias Kildetoft
Sep 6 at 9:31
2
2
You wrote yourself what $langle xrangle $ means just above it.
– Tobias Kildetoft
Sep 6 at 9:17
You wrote yourself what $langle xrangle $ means just above it.
– Tobias Kildetoft
Sep 6 at 9:17
Ok. What if $angle x^2+1angle$ means then ?
– blue boy
Sep 6 at 9:19
Ok. What if $angle x^2+1angle$ means then ?
– blue boy
Sep 6 at 9:19
2
2
Still, you have the definition right there, so I am not sure what you are really asking about.
– Tobias Kildetoft
Sep 6 at 9:21
Still, you have the definition right there, so I am not sure what you are really asking about.
– Tobias Kildetoft
Sep 6 at 9:21
@TobiasKildetoft $<x^2+1>=f(x) in R[x]$.
– blue boy
Sep 6 at 9:26
@TobiasKildetoft $<x^2+1>=f(x) in R[x]$.
– blue boy
Sep 6 at 9:26
2
2
You would make things more clear for yourself if you stop writing $f(x)$ for the polynomial $f$, since that makes it hard to tell the difference between evaluating $f$ at $x^2+1$ and multiplying $f$ by $x^2+1$.
– Tobias Kildetoft
Sep 6 at 9:31
You would make things more clear for yourself if you stop writing $f(x)$ for the polynomial $f$, since that makes it hard to tell the difference between evaluating $f$ at $x^2+1$ and multiplying $f$ by $x^2+1$.
– Tobias Kildetoft
Sep 6 at 9:31
 |Â
show 2 more comments
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
$langle xrangle$ is the set of all polynomials that can be written as a multiple of the polynomial $f(x)=x$. Let's take a look at your example:
Every polynomial $fin R[x]$ with $f(0)=0$ can be written as $f(x)=g(x)x$ for some $gin R[x]$. Vice versa, every polynomial of the form $f(x)=g(x)x$ satisfies $f(0)=0$. Therefore $langle xrangle = ;f(0)=0)$.
answered Sep 6 at 9:27
Janik
1,4352418
Ok. What if $angle x^2+1angle$ means then ?
– blue boy
Sep 6 at 9:30
1
It is the set of all polynomials that can be written as a multiple of the polynomial $x^2+1$. For example, $x^4+x^2=(x^2+1)x^2$ is such a polynomial, whereas for example $x$ or $x^2$ cannot be written in that form.
– Janik
Sep 6 at 9:33
I get it now . Thanks.
– blue boy
Sep 6 at 10:43
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
$langle xrangle$ is the set of all polynomials that can be written as a multiple of the polynomial $f(x)=x$. Let's take a look at your example:
Every polynomial $fin R[x]$ with $f(0)=0$ can be written as $f(x)=g(x)x$ for some $gin R[x]$. Vice versa, every polynomial of the form $f(x)=g(x)x$ satisfies $f(0)=0$. Therefore $langle xrangle = ;f(0)=0)$.
answered Sep 6 at 9:27
Janik
1,4352418
Ok. What if $angle x^2+1angle$ means then ?
– blue boy
Sep 6 at 9:30
1
It is the set of all polynomials that can be written as a multiple of the polynomial $x^2+1$. For example, $x^4+x^2=(x^2+1)x^2$ is such a polynomial, whereas for example $x$ or $x^2$ cannot be written in that form.
– Janik
Sep 6 at 9:33
I get it now . Thanks.
– blue boy
Sep 6 at 10:43
add a comment |Â
up vote
2
down vote
accepted
$langle xrangle$ is the set of all polynomials that can be written as a multiple of the polynomial $f(x)=x$. Let's take a look at your example:
Every polynomial $fin R[x]$ with $f(0)=0$ can be written as $f(x)=g(x)x$ for some $gin R[x]$. Vice versa, every polynomial of the form $f(x)=g(x)x$ satisfies $f(0)=0$. Therefore $langle xrangle = ;f(0)=0)$.
answered Sep 6 at 9:27
Janik
1,4352418
Ok. What if $angle x^2+1angle$ means then ?
– blue boy
Sep 6 at 9:30
1
It is the set of all polynomials that can be written as a multiple of the polynomial $x^2+1$. For example, $x^4+x^2=(x^2+1)x^2$ is such a polynomial, whereas for example $x$ or $x^2$ cannot be written in that form.
– Janik
Sep 6 at 9:33
I get it now . Thanks.
– blue boy
Sep 6 at 10:43
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
$langle xrangle$ is the set of all polynomials that can be written as a multiple of the polynomial $f(x)=x$. Let's take a look at your example:
Every polynomial $fin R[x]$ with $f(0)=0$ can be written as $f(x)=g(x)x$ for some $gin R[x]$. Vice versa, every polynomial of the form $f(x)=g(x)x$ satisfies $f(0)=0$. Therefore $langle xrangle = ;f(0)=0)$.
answered Sep 6 at 9:27
Janik
1,4352418
$langle xrangle$ is the set of all polynomials that can be written as a multiple of the polynomial $f(x)=x$. Let's take a look at your example:
Every polynomial $fin R[x]$ with $f(0)=0$ can be written as $f(x)=g(x)x$ for some $gin R[x]$. Vice versa, every polynomial of the form $f(x)=g(x)x$ satisfies $f(0)=0$. Therefore $langle xrangle = ;f(0)=0)$.
answered Sep 6 at 9:27
Janik
1,4352418
answered Sep 6 at 9:27
Janik
1,4352418
answered Sep 6 at 9:27
Janik
1,4352418
answered Sep 6 at 9:27
Janik
1,4352418
1,4352418
Ok. What if $angle x^2+1angle$ means then ?
– blue boy
Sep 6 at 9:30
1
It is the set of all polynomials that can be written as a multiple of the polynomial $x^2+1$. For example, $x^4+x^2=(x^2+1)x^2$ is such a polynomial, whereas for example $x$ or $x^2$ cannot be written in that form.
– Janik
Sep 6 at 9:33
I get it now . Thanks.
– blue boy
Sep 6 at 10:43
add a comment |Â
Ok. What if $angle x^2+1angle$ means then ?
– blue boy
Sep 6 at 9:30
1
It is the set of all polynomials that can be written as a multiple of the polynomial $x^2+1$. For example, $x^4+x^2=(x^2+1)x^2$ is such a polynomial, whereas for example $x$ or $x^2$ cannot be written in that form.
– Janik
Sep 6 at 9:33
I get it now . Thanks.
– blue boy
Sep 6 at 10:43
Ok. What if $angle x^2+1angle$ means then ?
– blue boy
Sep 6 at 9:30
Ok. What if $angle x^2+1angle$ means then ?
– blue boy
Sep 6 at 9:30
1
1
It is the set of all polynomials that can be written as a multiple of the polynomial $x^2+1$. For example, $x^4+x^2=(x^2+1)x^2$ is such a polynomial, whereas for example $x$ or $x^2$ cannot be written in that form.
– Janik
Sep 6 at 9:33
It is the set of all polynomials that can be written as a multiple of the polynomial $x^2+1$. For example, $x^4+x^2=(x^2+1)x^2$ is such a polynomial, whereas for example $x$ or $x^2$ cannot be written in that form.
– Janik
Sep 6 at 9:33
I get it now . Thanks.
– blue boy
Sep 6 at 10:43
I get it now . Thanks.
– blue boy
Sep 6 at 10:43
add a comment |Â
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2
You wrote yourself what $langle xrangle $ means just above it.
– Tobias Kildetoft
Sep 6 at 9:17
Ok. What if $angle x^2+1angle$ means then ?
– blue boy
Sep 6 at 9:19
2
Still, you have the definition right there, so I am not sure what you are really asking about.
– Tobias Kildetoft
Sep 6 at 9:21
@TobiasKildetoft $<x^2+1>=f(x) in R[x]$.
– blue boy
Sep 6 at 9:26
2
You would make things more clear for yourself if you stop writing $f(x)$ for the polynomial $f$, since that makes it hard to tell the difference between evaluating $f$ at $x^2+1$ and multiplying $f$ by $x^2+1$.
– Tobias Kildetoft
Sep 6 at 9:31