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In this paper we first give elementwise characterizations of real maximal ideals and maximal  z °-ideals of  C ( X ). Next, using this we characterize the real maximal ideals of the classical ring of quotients  q ( X ) of  C ( X ) and also maximal  z °-ideals of factor rings of  C ( X ). We show that every real maximal ideal of  q ( X ) is precisely the extension of a real maximal ideal of  C ( X ) which is also a  z ° -ideal, i.e., the extension of  M pfor some almost  P  -point  p  of  υX . Using this fact it turns out that in contrast to  C ( X ),  q ( X ) may not contain any real maximal ideal. We observe that every maximal ideal of  q ( X ) is real if and only if  X  is a pseudocompact almost  P  -space, i.e.,  q ( X ) =  C* ( X ). We also observe that the real maximal ideals of  q ( X ) are precisely the extensions of the real maximal ideals of  C ( X ) when and only when  X  is an almost  P  -space, i.e.,  q ( X ) =  C ( X ).

Characterizations of maximal z °-ideals of C ( X ) and real maximal ideals of q ( X )