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

problem:

Determine all ideals of Z/Z10.

solution:



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?

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.

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 of a ring then the set of ideals is the set . In particular, in a (or more generally a PID) this says that the ideals of are of the form where , or equivalently . From this it's easy to deduce that the number of ideals of is the number of (positive) divisors of . So, as a special case .

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 have the property that 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.

Thanks. I appreciate it.