On C ( X ) Modulo its Socle

What is it about?

Let CF (X) denote the socle of C(X). It is shown that X is a P-space if and only if C(X) is a א0 -self-injective ring or equivalently, if and only if C(X)/CF (X) is א0 -self-injective. We also prove that X is an extremally disconnected P-space with only a finite number of isolated points if and only if C(X)/CF (X) is self-injective. Consequently, if X is a P-space, then X is either an extremally disconnected space with at most a countable number of isolated points or both C(X) and C(X)/CF (X) have uncountable Goldie-dimensions. Prime ideals of C(X)/CF (X) are also studied.

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Why is it important?

It is well-known that commutative regular rings (even with some strong properties) are far away from being א0 -self-injective. In this article showed that the regularity of C(X) coincides with its א0 -self-injective.

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