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Basis of module and Free module

  1. Mar 5, 2014 #1
    We define basis as:
    Let M be a module over a ring R with unity and let S be a subset of M. Then S is called a basis of M if
    1. M is generated by S 2. S is linearly independent set.

    Also we define free module as
    An R module M is called a free module if there exists a subset S of M s.t.S generates M and S is linearly independent set.
    NOW my QUESTIONS are :
    1.Does a free module require unity?
    2.Can a free module be basis?
    3.Please tell me also the difference between the notations {0},{},(0) in reference to modules.

    * I am still a learner in this area of mathematics.
     
  2. jcsd
  3. Mar 5, 2014 #2

    micromass

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    You mean whether the underlying ring ##R## needs to have a unit? No, that's not necessary to define free module.

    No, a free module always has a basis, but it never is a basis. A free module is the same as a vector space over a field. A vector space always has a basis, but I'm sure you know a vector space never is a basis.

    The set ##\{0\}## is the set with as only element ##0##.
    The set ##\{\}## is the empty set ##\emptyset##.
    The set ##(0)## is the ideal generated by the element ##0##. It coincides with ##\{0\}##.
     
  4. Mar 11, 2014 #3
    Search about "free objects" in a category (in category theory).
     
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