Math 361, Spring 2014, Assignment 13

Carefully define the following terms, then give one example and one non-example of each:

1. Group presentation.
2. Generators (of a given presentation).
3. Relators (of a presentation).
4. Finitely presented group.
5. Isomorphic presentations.
6. Irreducible element (of a domain $D$).
7. Prime element (of a domain $D$).

Carefully state the following theorems (you need not prove them):

1. Theorem relating unique factorization to divisor chains and primeness.
2. Theorem relating unique factorization in $D$ to unique factorization in $D[x]$.
3. Theorem concerning factorization in principal ideal domains.

Solve the following problems:

1. Section 40, problems 1, 3, and 5.
2. Section 45, problems 1, 3, 5, 7, and 9.

--------------------End of assignment--------------------

Questions:

I am not entirely sure, but I think the first theorem was essentially stated in two parts in lecture. We showed that if we fix an integral domain $D$, and $D$ satisfies the DCC, then any element can be expressed as the product of irreducibles (i.e. existence of factorization). Also, if $D$ satisfies the primeness condition, I believe it then admits unique factorization. So then the theorem would be that any integral domain satisfying the DCC and the primeness condition is a UFD. I am not sure whether this goes both ways (i.e. if being a UFD implies DCC and primeness condition). --Robert.Moray (talk) 10:11, 5 May 2014 (EDT)

We only proved it in one direction, but it does in fact go both ways: a domain $D$ is a UFD if and only if it satisfies the divisor chain condition (every divisor chain eventually stabilizes) and the primeness condition (every irreducible element is prime). --Steven.Jackson (talk) 16:14, 7 May 2014 (EDT)