On the classical ring of quotients of c ( x )

Qcl (X) and Q(X) denote the classical and maximal rings of quotients of C (X) respectively. We investigate certain algebraic properties of the ring, Qcl (X) for a Tychonoff space X , including z-ideals, minimal ideals and the socle of the ring. We provide several characterisations of Qcl (X) as a Von-Neumann regular ring. We define nowhere almost P -spaces using the ring Qcl (X), and in the class of Tychonoff spaces, we characterise them as spaces with no non-isolated almost P -points. We show that a Tychonoff space with countable pseudocharacter is a nowhere almost P -space and highlight that this condition is not superfluous using the closed ordinal space. We also note that the socle of Qcl (X) is free if and only if the space X is a nowhere almost P -space.