Abstract Algebra: a problem about ideal

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Homework Statement


Let J be the set of all polynomials with zero constant term in Z[x]. (Z=integers)
a.) Show that J is the principal ideal (x) in Z[x].
b.) Show that Z[x]/J consists of an infinite number of distinct cosets, one for each n[tex]\in[/tex]Z.


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The Attempt at a Solution


I have trouble understanding what a principal ideal is. Any help on how I should start would be great. Thanks!
 

Answers and Replies

  • #2
Dick
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Well, what's the set generated by {x} in Z(x)? Isn't it just x*Z(x)? What's that?
 
  • #3
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An ideal is a subset of a ring that's closed under addition of its own elements and under multiplication by anything in the ring. A principal ideal is one that's generated by one element.

So, take an element in Z[x] and multiply it by x. Whatever you get, by definition, is in the ideal (x). What do those things look like?

Now, two things are in the same coset of Z[x]/J if their difference is in J. So, if I can subtract off any polynomial I want with zero constant term, what stays unchanged?
 

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