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Let  X  be a topological space and let    be an ideal of subsets of  X . The space  X  is called  connected moduloif there is no continuous mapping  f  :  X  → [0, 1] which is 2 -valued moduloin the sense that neither  f−1 (0) nor  f−1 (1) belongs to    but  X  \ ( f−1 (0) ∪  f−1 (1)) belongs to   . We prove that a completely regular space  X  is connected modulo    if and only if the quotient    of the ring  C B( X ) (of all bounded continuous real-valued mappings on  X  equipped with pointwise addition and multiplication) is indecomposable. Here    is the ideal of  C B( X ) consisting of all  f  in  C B( X ) such that | f | −1 ([ ε , ∞)) belongs to    for any positive  ε . We examine examples corresponding to various choices of the ideal   . We conclude with consideration of the ideal    of  CB  ( X ) whose importance is highlighted by our characterization theorem.

Connectedness modulo an ideal; a characterization theorem