zero moduleの例文

• A trivial algebra over a field is simultaneously a zero vector space considered algebra is simultaneously a zero module.
• An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules.
• If, for some, the th syzygy module is free, then, by taking a basis as a generating set, the next syzygy module ( and every subsequent one ) is the zero module.
• Some authors work only over integral domains and use this condition as the definition of a torsion-free module, but this does not work well over more general rings, for if the ring contains zero divisors then the only module satisfying this condition is the zero module.
• As the relations form a module, one may consider the relations between the relations; Hilbert's syzygy theorem asserts that, if one continues in this way, starting with a module over a polynomial ring in indeterminates over a field, one eventually finds a zero module of relations, after at most steps.