# Abstract Algebra: a problem about ideal

1. Mar 8, 2010

### iwonde

1. The problem statement, all variables and given/known data
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$$\in$$Z.

2. Relevant equations

3. 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!

2. Mar 8, 2010

### Dick

Well, what's the set generated by {x} in Z(x)? Isn't it just x*Z(x)? What's that?

3. Mar 8, 2010

### Tinyboss

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?