Skip to main content
Bumped by Community user
Bumped by Community user
Bumped by Community user
Bumped by Community user

Let $R$ be a commutative reduced ring with ideantityidentity with the property that if $I$ and $J$ are two ideals of $R $ such that if $I+J $$I+J$ is not contained in any minimal prime ideal, then there exist ideals $I' $$I'$ and $J' $$J'$ of $R$ such that $ I'J'=0 $$I'J'=0$ and the ideals $I+I' $$I+I'$ and $J+J' $$J+J'$ are not contained in any minimal prime ideal.

Is there any charactrizationcharacterization for such a ring? Or

Is there a reduced ring that dose not have this property?

Note that clearly noetherianNoetherian rings have this property.

Let $R$ be a commutative reduced ring with ideantity with the property that if $I$ and $J$ are two ideals of $R $ such that if $I+J $ is not contained in any minimal prime ideal, then there exist ideals $I' $ and $J' $ of $R$ such that $ I'J'=0 $ and the ideals $I+I' $ and $J+J' $ are not contained in any minimal prime ideal.

Is there any charactrization for such a ring? Or

Is there a reduced ring that dose not have this property?

Note that clearly noetherian rings have this property.

Let $R$ be a commutative reduced ring with identity with the property that if $I$ and $J$ are two ideals of $R $ such that if $I+J$ is not contained in any minimal prime ideal, then there exist ideals $I'$ and $J'$ of $R$ such that $I'J'=0$ and the ideals $I+I'$ and $J+J'$ are not contained in any minimal prime ideal.

Is there any characterization for such a ring? Or

Is there a reduced ring that dose not have this property?

Note that clearly Noetherian rings have this property.

Bumped by Community user
typo in title
Link
Gerry Myerson
  • 39.9k
  • 10
  • 186
  • 247

A relation between ideasideals and annihilators

added 63 characters in body
Source Link

Let $R$ be a commutative reduced ring with ideantity with the property that if $I$ and $J$ are two ideals of $R $ such that if $I+J $ is not contained in any minimal prime ideal, then there exist ideals $I' $ and $J' $ of $R$ such that $ I'J'=0 $ and the ideals $I+I' $ and $J+J' $ are not contained in any minimal prime ideal.

Is there any charactrization for such a ring? Or

Is there a reduced ring that dose not have this property?

Note that clearly noetherian rings have this property.

Let $R$ be a commutative reduced ring with ideantity with the property that if $I$ and $J$ are two ideals of $R $ such that if $I+J $ is not contained in any minimal prime ideal, then there exist ideals $I' $ and $J' $ of $R$ such that $ I'J'=0 $ and the ideals $I+I' $ and $J+J' $ are not contained in any minimal prime ideal.

Is there any charactrization for such a ring?

Note that clearly noetherian rings have this property.

Let $R$ be a commutative reduced ring with ideantity with the property that if $I$ and $J$ are two ideals of $R $ such that if $I+J $ is not contained in any minimal prime ideal, then there exist ideals $I' $ and $J' $ of $R$ such that $ I'J'=0 $ and the ideals $I+I' $ and $J+J' $ are not contained in any minimal prime ideal.

Is there any charactrization for such a ring? Or

Is there a reduced ring that dose not have this property?

Note that clearly noetherian rings have this property.

Source Link
Loading