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

• Dec 5th 2011, 03:52 PM
Gantz
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?
• Dec 5th 2011, 09:46 PM
Drexel28
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.
• Dec 6th 2011, 08:11 AM
Gantz
Re: Understanding a solution -- intro to ring theory -- ideals
Thanks. I appreciate it.
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 15, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}