Mixing
$I$ an irreducible ideal in a PID is a prime ideal
$begingroup$
$I$ is irreducible ideal of $R$ when $R$ is a PID , I want to show it's a prime ideal.
The definition I am familiar with: $I$ a proper ideal of $R$ is an irreducible ideal if for any two ideals $J,K$ such that $Isubseteq J$, $Isubseteq K$ , then $Ineq Jcap K$
Well denoting $I=langle arangle$ , Assuming $I$ is not a primes, means there are $x,yin R$ , such that $xyinlangle arangle$ such that $xnotinlangle a rangle$ , $ynotinlangle a rangle$.
Taking $langle x rangle cap langle y rangle$ , we get an ideal such that $ain langle x rangle cap langle y rangle$ because $a=xyin langle x rangle$ , $a=xyin langle y rangle$. So $langle a rangle subsetlangle x ranglecap langle y rangle$.
For $rin langle x ranglecap langle y rangle$
I tried to show $rin langle a rangle$ but didn't succeed, I gotthe next results:
- $r = xd , r=ye Rightarrow r^2=xyed=acdot edRightarrow r^2 in langle a rangle$
$r=xd , r=ye Rightarrow ry=xyd=ad$ and$ rx=yxe=aeRightarrow rx,ryin langle a rangle$.
abstract-algebra ring-theory ideals principal-ideal-domains
asked Jan 24 at 18:08
dandan
607613
$endgroup$
add a comment |
$begingroup$
$I$ is irreducible ideal of $R$ when $R$ is a PID , I want to show it's a prime ideal.
The definition I am familiar with: $I$ a proper ideal of $R$ is an irreducible ideal if for any two ideals $J,K$ such that $Isubseteq J$, $Isubseteq K$ , then $Ineq Jcap K$
Well denoting $I=langle arangle$ , Assuming $I$ is not a primes, means there are $x,yin R$ , such that $xyinlangle arangle$ such that $xnotinlangle a rangle$ , $ynotinlangle a rangle$.
Taking $langle x rangle cap langle y rangle$ , we get an ideal such that $ain langle x rangle cap langle y rangle$ because $a=xyin langle x rangle$ , $a=xyin langle y rangle$. So $langle a rangle subsetlangle x ranglecap langle y rangle$.
For $rin langle x ranglecap langle y rangle$
I tried to show $rin langle a rangle$ but didn't succeed, I gotthe next results:
- $r = xd , r=ye Rightarrow r^2=xyed=acdot edRightarrow r^2 in langle a rangle$
$r=xd , r=ye Rightarrow ry=xyd=ad$ and$ rx=yxe=aeRightarrow rx,ryin langle a rangle$.
abstract-algebra ring-theory ideals principal-ideal-domains
asked Jan 24 at 18:08
dandan
607613
$endgroup$
$begingroup$
The question in the link talks about $ain$PID is irreducible $Leftrightarrow$ $a$ is prime. Yet I am asking about an irreducible Ideal, not about an irreducible element. Can someone please explain the equivalence (if exists)?
$endgroup$
– dan
Jan 25 at 4:37
add a comment |
$begingroup$
$I$ is irreducible ideal of $R$ when $R$ is a PID , I want to show it's a prime ideal.
The definition I am familiar with: $I$ a proper ideal of $R$ is an irreducible ideal if for any two ideals $J,K$ such that $Isubseteq J$, $Isubseteq K$ , then $Ineq Jcap K$
Well denoting $I=langle arangle$ , Assuming $I$ is not a primes, means there are $x,yin R$ , such that $xyinlangle arangle$ such that $xnotinlangle a rangle$ , $ynotinlangle a rangle$.
Taking $langle x rangle cap langle y rangle$ , we get an ideal such that $ain langle x rangle cap langle y rangle$ because $a=xyin langle x rangle$ , $a=xyin langle y rangle$. So $langle a rangle subsetlangle x ranglecap langle y rangle$.
For $rin langle x ranglecap langle y rangle$
I tried to show $rin langle a rangle$ but didn't succeed, I gotthe next results:
- $r = xd , r=ye Rightarrow r^2=xyed=acdot edRightarrow r^2 in langle a rangle$
$r=xd , r=ye Rightarrow ry=xyd=ad$ and$ rx=yxe=aeRightarrow rx,ryin langle a rangle$.
abstract-algebra ring-theory ideals principal-ideal-domains
asked Jan 24 at 18:08
dandan
607613
$endgroup$
$I$ is irreducible ideal of $R$ when $R$ is a PID , I want to show it's a prime ideal.
The definition I am familiar with: $I$ a proper ideal of $R$ is an irreducible ideal if for any two ideals $J,K$ such that $Isubseteq J$, $Isubseteq K$ , then $Ineq Jcap K$
Well denoting $I=langle arangle$ , Assuming $I$ is not a primes, means there are $x,yin R$ , such that $xyinlangle arangle$ such that $xnotinlangle a rangle$ , $ynotinlangle a rangle$.
Taking $langle x rangle cap langle y rangle$ , we get an ideal such that $ain langle x rangle cap langle y rangle$ because $a=xyin langle x rangle$ , $a=xyin langle y rangle$. So $langle a rangle subsetlangle x ranglecap langle y rangle$.
For $rin langle x ranglecap langle y rangle$
I tried to show $rin langle a rangle$ but didn't succeed, I gotthe next results:
- $r = xd , r=ye Rightarrow r^2=xyed=acdot edRightarrow r^2 in langle a rangle$
$r=xd , r=ye Rightarrow ry=xyd=ad$ and$ rx=yxe=aeRightarrow rx,ryin langle a rangle$.
abstract-algebra ring-theory ideals principal-ideal-domains
abstract-algebra ring-theory ideals principal-ideal-domains
asked Jan 24 at 18:08
dandan
607613
asked Jan 24 at 18:08
dandan
607613
edited Jan 25 at 10:02
dan
asked Jan 24 at 18:08
dandan
607613
asked Jan 24 at 18:08
dandan
607613
asked Jan 24 at 18:08
dandan
607613
607613
$begingroup$
The question in the link talks about $ain$PID is irreducible $Leftrightarrow$ $a$ is prime. Yet I am asking about an irreducible Ideal, not about an irreducible element. Can someone please explain the equivalence (if exists)?
$endgroup$
– dan
Jan 25 at 4:37
add a comment |
$begingroup$
The question in the link talks about $ain$PID is irreducible $Leftrightarrow$ $a$ is prime. Yet I am asking about an irreducible Ideal, not about an irreducible element. Can someone please explain the equivalence (if exists)?
$endgroup$
– dan
Jan 25 at 4:37
$begingroup$
The question in the link talks about $ain$PID is irreducible $Leftrightarrow$ $a$ is prime. Yet I am asking about an irreducible Ideal, not about an irreducible element. Can someone please explain the equivalence (if exists)?
$endgroup$
– dan
Jan 25 at 4:37
$begingroup$
The question in the link talks about $ain$PID is irreducible $Leftrightarrow$ $a$ is prime. Yet I am asking about an irreducible Ideal, not about an irreducible element. Can someone please explain the equivalence (if exists)?
$endgroup$
– dan
Jan 25 at 4:37
add a comment |
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$begingroup$
The question in the link talks about $ain$PID is irreducible $Leftrightarrow$ $a$ is prime. Yet I am asking about an irreducible Ideal, not about an irreducible element. Can someone please explain the equivalence (if exists)?
$endgroup$
– dan
Jan 25 at 4:37