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Keywords:
modular lattice; essential ideal; max-semicomplement; extending ideal; direct summand; exchangeable decomposition; ojective ideal
modular lattice; essential ideal; max-semicomplement; extending ideal; direct summand; exchangeable decomposition; ojective ideal
Summary:
The concept of an extending ideal in a modular lattice is introduced. A translation of module-theoretical concept of ojectivity (i.e. generalized relative injectivity) in the context of the lattice of ideals of a modular lattice is introduced. In a modular lattice satisfying a certain condition, a characterization is given for direct summands of an extending ideal to be mutually ojective. We define exchangeable decomposition and internal exchange property of an ideal in a modular lattice. It is shown that a finite decomposition of an extending ideal is exchangeable if and only if its summands are mutually ojective.
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