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Examples ofcoefficient ring
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The next simplest case is the case when thecoefficientringis a principal ideal domain.
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This example is from Wikipedia and may be reused under a CC BY-SA license.As thecoefficientringbecomes more general, the relationship becomes more complicated.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Here, homology and cohomology is taken with coefficients in the ring of integers, but the isomorphism holds for anycoefficientring.
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This example is from Wikipedia and may be reused under a CC BY-SA license.A square matrix has an inverse if and only if its determinant has an inverse in thecoefficientring.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The stable homotopy groups form thecoefficientringof an extraordinary cohomology theory, called stable cohomotopy theory.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The cobordism classes of manifolds form a ring that is usually thecoefficientringof some generalized cohomology theory.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Conversely, if a finitely generated algebra is integral (over thecoefficientring), then it is finitely generated module.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The proof is valid for arbitrary commutative coefficient rings.
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This example is from Wikipedia and may be reused under a CC BY-SA license.These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors.
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