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Sándor Kovács
Sándor Kovács
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Let P be the property of "being an integral domain". Then

1: If $A$ is an integral domain, then $A_p$ is an integral domain for every prime ideal $p\subseteq A$.

On the other hand.

2: Let $A=A_1\oplus A_2$ be a direct sum of two integral domains. Then it is obviously not an integral domain, although $A_p$ is an integral domain for every prime ideal $p\subseteq A$. So P is not local.

Let P be the property of "being an integral domain". Then

1: If $A$ is an integral domain, then $A_p$ is an integral domain for every prime ideal $p\subseteq A$.

On the other hand.

2: Let $A=A_1\oplus A_2$ be a direct sum of two integral domains. Then it is obviously not an integral domain, although $A_p$ is an integral domain for every prime ideal $p\subseteq A$.

Let P be the property of "being an integral domain". Then

1: If $A$ is an integral domain, then $A_p$ is an integral domain for every prime ideal $p\subseteq A$.

On the other hand.

2: Let $A=A_1\oplus A_2$ be a direct sum of two integral domains. Then it is obviously not an integral domain, although $A_p$ is an integral domain for every prime ideal $p\subseteq A$. So P is not local.

Source Link
Sándor Kovács
Sándor Kovács
  • 42.9k
  • 2
  • 109
  • 155

Let P be the property of "being an integral domain". Then

1: If $A$ is an integral domain, then $A_p$ is an integral domain for every prime ideal $p\subseteq A$.

On the other hand.

2: Let $A=A_1\oplus A_2$ be a direct sum of two integral domains. Then it is obviously not an integral domain, although $A_p$ is an integral domain for every prime ideal $p\subseteq A$.