P-space

Suppose $X$ is a completely regular topological space. Then $X$ is said to be a P-space if every prime ideal in $C(X)$ , the ring of continuous functions on $X$ , is maximal.

For example, every space with the discrete topology is a P-space.

Algebraically, a commutative reduced ring $R$ with $1$ such that every prime ideal is maximal is equivalent to any of the following statements:

•
$R$ is von-Neumann regular,
•
every ideal in $R$ is the intersection of prime ideals,
•
every ideal in $R$ is the intersection of maximal ideals,
•
every principal ideal is generated by an idempotent.

When $R=C(X)$ , then $R$ is commutative reduced with $1$ . In addition to the algebraic characterizations of $R$ above, $X$ being a P-space is equivalent to any of the following statements:

•
every zero set is open
•
if $f,g\\in C(X)$ , then $(f,g)=(f^{2}+g^{2})$ .

Some properties of P-spaces:

1.
Every subspace of a P-space is a P-space,
2.
Every quotient space of a P-space is a P-space,
3.
Every finite product of P-spaces is a P-space,
4.
Every P-space has a base of clopen sets.

For more properties of P-spaces, please see the reference below. For proofs of the above properties and equivalent characterizations, see http://planetmath.org/?op=getobj&from=books&id=46here.

References

1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).

单词	Pspace
释义	P-space Suppose $X$ is a completely regular topological space. Then $X$ is said to be a P-space if every prime ideal in $C(X)$ , the ring of continuous functions on $X$ , is maximal. For example, every space with the discrete topology is a P-space. Algebraically, a commutative reduced ring $R$ with $1$ such that every prime ideal is maximal is equivalent to any of the following statements: • $R$ is von-Neumann regular, • every ideal in $R$ is the intersection of prime ideals, • every ideal in $R$ is the intersection of maximal ideals, • every principal ideal is generated by an idempotent. When $R=C(X)$ , then $R$ is commutative reduced with $1$ . In addition to the algebraic characterizations of $R$ above, $X$ being a P-space is equivalent to any of the following statements: • every zero set is open • if $f,g\\in C(X)$ , then $(f,g)=(f^{2}+g^{2})$ . Some properties of P-spaces: 1. Every subspace of a P-space is a P-space, 2. Every quotient space of a P-space is a P-space, 3. Every finite product of P-spaces is a P-space, 4. Every P-space has a base of clopen sets. For more properties of P-spaces, please see the reference below. For proofs of the above properties and equivalent characterizations, see http://planetmath.org/?op=getobj&from=books&id=46here. References 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
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