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Given an ideal I on the positive integers, a real sequence (x n) is said to be I -statistically convergent to ℓ provided that n ∈ N : 1 n | { k ≤ n : x k ∉ U } | ≥ ε ∈ I for all neighborhoods U of ℓ and all ε > 0. First, we show that I -statistical convergence coincides with J -convergence, for some unique ideal J = J (I). In addition, J is Borel [analytic, coanalytic, respectively] whenever I is Borel [analytic, coanalytic, resp.]. Then we prove, among others, that if I is the summable ideal { A ⊆ N : ∑ a ∈ A 1 ∕ a < ∞ } or the density zero ideal { A ⊆ N : lim n → ∞ 1 n | A ∩ [ 1 , n ] | = 0 } then I -statistical convergence coincides with statistical convergence. This can be seen as a Tauberian theorem which extends a classical theorem of Fridy. Lastly, we show that this is never the case if I is maximal. [ABSTRACT FROM AUTHOR]

It is well known that the set C[0, 1] of real-valued continuous functions on the closed interval [0, 1] has a natural ring structure. Its maximal ideals are known to be points. Here we show that although there exist prime ideals which are not maximal, the zero set is again a singleton. In particular, each prime ideal is contained in a unique maximal ideal. [ABSTRACT FROM AUTHOR]

We generalize the Bolzano-Weierstrass theorem (that every bounded sequence of reals admits a convergent subsequence) on ideal convergence. We show examples of ideals with and without the Bolzano-Weierstrass property, and give characterizations of BW property in terms of submeasures and extendability to a maximal P-ideal. We show applications to Rudin-Keisler and Rudin-Blass orderings of ideals and quotient Boolean algebras. In particular we show that an ideal does not have BW property if and only if its quotient Boolean algebra has a countably splitting family. [ABSTRACT FROM AUTHOR]