in this paper, we introduce a method by which we can find a close connection between the set of prime $z$-ideals of $c(x)$ and the same of $c(y)$, for some special subset $y$ of $x$. for instance, if $y=coz(f)$ for some $fin c(x)$, then there exists a one-to-one correspondence between the set of prime $z$-ideals of $c(y)$ and the set of prime $z$-ideals of $c(x)$ not containing $f$. moreover, c...

We derive transitivity of various degrees of proximinality in Banach spaces. When the transitivity does not carry forward to the bigger space we investigate these properties under some additional assumptions of the intermediate space. For instance, we show that if Z ⊆ Y ⊆ X where Z is a finite co-dimensional subspace of X which is strongly proximinal in Y and Y is an M-ideal in X then Z is stro...

We give a new constructive definition for Noetherian rings. It has a very concrete statement and is nevertheless strong enough to prove constructively the termination of algorithms involving “trees of ideals”. The efficiency of such algorithms (at least for providing clear and intuitive constructive proofs) is illustrated in a section about Lasker–Noether rings: we give constructive proofs for ...

A set X ⊆ R is strongly meager if for every measure zero set H, X + H = R. Let SM denote the collection of strongly meager sets. We show that assuming CH, SM is not an ideal.

A subset selection Z assigns to each partially ordered set P a certain collection ZP of subsets. The theory of topological and of algebraic (i.e. finitary) closure spaces extends to the general Z-level, by replacing finite or directed sets, respectively, with arbitrary ‘Z-sets’. This leads to a theory of Z-union completeness, Z-arity, Z-soberness etc. Order-theoretical notions such as complete ...