Is the annihilator of a minimal prime always principal in a reduced ring?
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$DeclareMathOperator{Ann}{Ann}$Let $R$ be a (one-dimensional) reduced ring and let $I$ be a non-zero minimal prime ideal of $R$.
I am looking for an example such that $Ann(I)$ is not principal.
Background:
This question arose when I was asking myself whether $R/I$ is isomorphic to $Ann(I)$ as $R/I$-modules. And that question arose when I was asking myself:
Where is the difference between making an $R$-module $M$ into an $R/I$-module via
- tensoring with $R/I$ over $R$, or
- via tensoring with $Ann(I)$ over $R$?
Thank you in advance!
abstract-algebra algebraic-geometry ring-theory commutative-algebra maximal-and-prime-ideals
asked Nov 13 at 9:45
windsheaf
590312
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up vote
0
down vote
favorite
$DeclareMathOperator{Ann}{Ann}$Let $R$ be a (one-dimensional) reduced ring and let $I$ be a non-zero minimal prime ideal of $R$.
I am looking for an example such that $Ann(I)$ is not principal.
Background:
This question arose when I was asking myself whether $R/I$ is isomorphic to $Ann(I)$ as $R/I$-modules. And that question arose when I was asking myself:
Where is the difference between making an $R$-module $M$ into an $R/I$-module via
- tensoring with $R/I$ over $R$, or
- via tensoring with $Ann(I)$ over $R$?
Thank you in advance!
abstract-algebra algebraic-geometry ring-theory commutative-algebra maximal-and-prime-ideals
asked Nov 13 at 9:45
windsheaf
590312
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
$DeclareMathOperator{Ann}{Ann}$Let $R$ be a (one-dimensional) reduced ring and let $I$ be a non-zero minimal prime ideal of $R$.
I am looking for an example such that $Ann(I)$ is not principal.
Background:
This question arose when I was asking myself whether $R/I$ is isomorphic to $Ann(I)$ as $R/I$-modules. And that question arose when I was asking myself:
Where is the difference between making an $R$-module $M$ into an $R/I$-module via
- tensoring with $R/I$ over $R$, or
- via tensoring with $Ann(I)$ over $R$?
Thank you in advance!
abstract-algebra algebraic-geometry ring-theory commutative-algebra maximal-and-prime-ideals
asked Nov 13 at 9:45
windsheaf
590312
$DeclareMathOperator{Ann}{Ann}$Let $R$ be a (one-dimensional) reduced ring and let $I$ be a non-zero minimal prime ideal of $R$.
I am looking for an example such that $Ann(I)$ is not principal.
Background:
This question arose when I was asking myself whether $R/I$ is isomorphic to $Ann(I)$ as $R/I$-modules. And that question arose when I was asking myself:
Where is the difference between making an $R$-module $M$ into an $R/I$-module via
- tensoring with $R/I$ over $R$, or
- via tensoring with $Ann(I)$ over $R$?
Thank you in advance!
abstract-algebra algebraic-geometry ring-theory commutative-algebra maximal-and-prime-ideals
abstract-algebra algebraic-geometry ring-theory commutative-algebra maximal-and-prime-ideals
asked Nov 13 at 9:45
windsheaf
590312
asked Nov 13 at 9:45
windsheaf
590312
asked Nov 13 at 9:45
windsheaf
590312
asked Nov 13 at 9:45
windsheaf
590312
asked Nov 13 at 9:45
windsheaf
590312
590312
add a comment |
add a comment |
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