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This is a follow up of the question https://math.stackexchange.com/questions/1189814/example-of-a-finitely-generated-faithful-torsion-module-over-a-commutative-ringExample of a finitely generated faithful torsion module over a commutative ring on MathSE.

Let $M$ be a finitely generated module over a commutative ring $R$ with the property that $\operatorname{Ann}_R x\ne 0$ for all $x \in M$. When $\operatorname{Ann}_R M\ne0$?

The simplest case is $R$ an integral domain. But what about $R$ (local) artinian, or noetherian? (In the counterexample I gave to the linked question $R$ is a commutative ring which is not noetherian.)

This is a follow up of https://math.stackexchange.com/questions/1189814/example-of-a-finitely-generated-faithful-torsion-module-over-a-commutative-ring.

Let $M$ be a finitely generated module over a commutative ring $R$ with the property that $\operatorname{Ann}_R x\ne 0$ for all $x \in M$. When $\operatorname{Ann}_R M\ne0$?

The simplest case is $R$ an integral domain. But what about $R$ (local) artinian, or noetherian? (In the counterexample I gave to the linked question $R$ is a commutative ring which is not noetherian.)

This is a follow up of the question Example of a finitely generated faithful torsion module over a commutative ring on MathSE.

Let $M$ be a finitely generated module over a commutative ring $R$ with the property that $\operatorname{Ann}_R x\ne 0$ for all $x \in M$. When $\operatorname{Ann}_R M\ne0$?

The simplest case is $R$ an integral domain. But what about $R$ (local) artinian, or noetherian? (In the counterexample I gave to the linked question $R$ is a commutative ring which is not noetherian.)

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This is a follow up of http://math.stackexchange.com/questions/1189814/example-of-a-finitely-generated-faithful-torsion-module-over-a-commutative-ringhttps://math.stackexchange.com/questions/1189814/example-of-a-finitely-generated-faithful-torsion-module-over-a-commutative-ring.

Let $M$ be a finitely generated module over a commutative ring $R$ with the property that $\operatorname{Ann}_R x\ne 0$ for all $x \in M$. When $\operatorname{Ann}_R M\ne0$?

The simplest case is $R$ an integral domain. But what about $R$ (local) artinian, or noetherian? (In the counterexample I gave to the linked question $R$ is a commutative ring which is not noetherian.)

This is a follow up of http://math.stackexchange.com/questions/1189814/example-of-a-finitely-generated-faithful-torsion-module-over-a-commutative-ring.

Let $M$ be a finitely generated module over a commutative ring $R$ with the property that $\operatorname{Ann}_R x\ne 0$ for all $x \in M$. When $\operatorname{Ann}_R M\ne0$?

The simplest case is $R$ an integral domain. But what about $R$ (local) artinian, or noetherian? (In the counterexample I gave to the linked question $R$ is a commutative ring which is not noetherian.)

This is a follow up of https://math.stackexchange.com/questions/1189814/example-of-a-finitely-generated-faithful-torsion-module-over-a-commutative-ring.

Let $M$ be a finitely generated module over a commutative ring $R$ with the property that $\operatorname{Ann}_R x\ne 0$ for all $x \in M$. When $\operatorname{Ann}_R M\ne0$?

The simplest case is $R$ an integral domain. But what about $R$ (local) artinian, or noetherian? (In the counterexample I gave to the linked question $R$ is a commutative ring which is not noetherian.)

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When finitely generateddoes a faithful modulesmodule have linearly independent elementsan element with zero annihilator?

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