Rings, fields, spaces etc.

  • Thread starter Gregg
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  • #1
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They all seem to be defined as sets with multiplication and addition axioms satisfied. What is the difference?
 

Answers and Replies

  • #2
737
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A field is a ring where every nonzero element has a multiplicative inverse. All fields are rings, but not vice-versa. What spaces are you talking about, vector spaces?
 
  • #3
459
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Maybe spaces is not accurate but there seem to be a lot of things which are defined as having satisfying similar axioms.
 
  • #4
737
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Yes, it's true, but they do all have their differences. Vector spaces, for example, need both a set of vectors and a field of scalars. You can treat a field as a vector space over itself, because of the similarity of the axioms, but they are intrinsically different.
 

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