Understanding a solution -- intro to ring theory -- ideals

problem:

Quote:

Determine all ideals of Z/Z10.

solution:
http://i1237.photobucket.com/albums/...lol/soln4a.jpg
http://i1237.photobucket.com/albums/...lol/soln4b.jpg

The first paragraph is just saying the ideals generated by the units in the ring is the whole ring correct?

Also, the principal ideals generated by 2, 4 and 8 are all the same correct? So those are three ideals or just one?
So, in the final answer there are 4 ideals in total right?

And finally why do we care about Ideals so much? What use do they even have besides giving students of average intelligence a giant headache in intro abstract algebra course lol?

Re: Understanding a solution -- intro to ring theory -- ideals

Quote:

Originally Posted by Gantz

The first paragraph is just saying the ideals generated by the units in the ring is the whole ring correct?

Right.

Quote:

Also, the principal ideals generated by 2, 4 and 8 are all the same correct? So those are three ideals or just one?
So, in the final answer there are 4 ideals in total right?

They are all the same ideal. There are four ideals. In general (this is the fourth isomorphism theorem) given an ideal $\mathfrak{a}$ of a ring $R$ then the set of ideals $R/\mathfrak{a}$ is the set $\{\mathfrak{b}/\mathfrak{a}:\mathfrak{b}\text{ is an ideal of }R\text{ and }\mathfrak{b}\supseteq\mathfrak{a}\}$ . In particular, in a $\mathbb{Z}$ (or more generally a PID) this says that the ideals of $\mathbb{Z}/(a)$ are of the form $(b)/(a)$ where $(b)\supseteq (a)$ , or equivalently $b\mid a$ . From this it's easy to deduce that the number of ideals of $\mathbb{Z}/(a)$ is the number $\sigma_0(a)$ of (positive) divisors of $a$ . So, as a special case $\sigma_0(10)=4$ .

Quote:

And finally why do we care about Ideals so much? What use do they even have besides giving students of average intelligence a giant headache in intro abstract algebra course lol?

Ideals encompass much of the theory of rings for multiple reasons. In one direction (the commutative algebraic direction) the ideals of a ring can function as "ideal" (get it now) elements of the ring. So, while a ring may not have unique factorization there may be a theory of unique factorization for its ideals.

In a different (more categorical) direction, we care mostly about rings up to isomorphism. In fact, what rings can be homomorphic images of a given ring, tell us a lot about the ring. Thus, we would like to know what rings $S$ have the property that $S$ is a homomorphic image of our ring. But, the first isomorphism gives us the answer to this question--the homomorphic images of a ring are precisely the quotient rings of our ring. It's clear then that quotient rings are of a huge importance in ring theory, and since one can define ideals as being precisely the (non-unital) subrings of a given ring for which quotient rings "make sense" ideals become equally important.

EDIT: I would suggest taking the time to read this.

Re: Understanding a solution -- intro to ring theory -- ideals

Thanks. I appreciate it.