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user26857
user26857
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The quoted result relies on the following elementary characterization of local regular rings:

Let $R$ be a local regular ring with maximal ideal $\mathfrak m$ and $x\in\mathfrak m-\mathfrak m^2$. Then $R$ is regular iff $R/(x)$ is regular and $x$ doesn't belong to any minimal prime.

The quoted result relies on the following elementary characterization of local regular rings:

Let $R$ be a local regular ring with maximal ideal $\mathfrak m$ and $x\in\mathfrak m-\mathfrak m^2$. Then $R$ is regular iff $R/(x)$ is regular and $x$ doesn't belong to any minimal prime.

The quoted result relies on the following elementary characterization of local regular rings:

Let $R$ be a local ring with maximal ideal $\mathfrak m$ and $x\in\mathfrak m-\mathfrak m^2$. Then $R$ is regular iff $R/(x)$ is regular and $x$ doesn't belong to any minimal prime.

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user26857
user26857
  • 1.3k
  • 14
  • 30

The quoted result relies on the following elementary characterization of local regular rings:

Let $R$ be a local regular ring with maximal ideal $\mathfrak m$ and $x\in\mathfrak m-\mathfrak m^2$. Then $R$ is regular iff $R/(x)$ is regular and $x$ doesn't belong to any minimal prime.