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Examples ofnilradical
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For purposes of comparison, consider thenilradicalof a commutative ring, which consists of all elements that are nilpotent.
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This example is from Wikipedia and may be reused under a CC BY-SA license.In the case of commutative rings, there is always a maximal nil ideal: thenilradicalof the ring.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Thenilradicalis equal to the intersection of all the ring's prime ideals.
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This example is from Wikipedia and may be reused under a CC BY-SA license.There are, however, analogues of thenilradicaldefined for noncommutative rings, that coincide with thenilradicalwhen commutativity is assumed.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The former condition ensures that thenilradicalof the ring is zero, so that the intersection of all the ring's minimal primes is zero.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The first example of a radical was thenilradicalintroduced in, based on a suggestion in.
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Wikipedia
This example is from Wikipedia and may be reused under a CC BY-SA license.This ideal is thenilradicalof the ring.
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Wikipedia
This example is from Wikipedia and may be reused under a CC BY-SA license.The singular radical contains the prime radical (thenilradicalin the case of commutative rings) but may properly contain it, even in the commutative case.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The set of all nilpotent elements itself need not be an ideal (or even a subgroup), so the uppernilradicalcan be much smaller than this set.
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Wikipedia
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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- be lost on someone
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- be lost without someone/something
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