Math 361, Spring 2015, Assignment 3
From cartan.math.umb.edu
Carefully define the following terms, then give one example and one non-example of each:[edit]
- Associate (of an element $a$ in a domain $D$).
Solve the following problems:[edit]
- Suppose that $a$ and $b$ are associate elements of a domain $D$. Prove that $a$ is irreducible if and only if $b$ is irreducible.
- Suppose that $a$ and $b$ are associate elements of a domain $D$. Prove that the set of multiples of $a$ is equal to the set of multiples of $b$. (The set of multiples of a given element is called the principal ideal generated by that element, and it will play a large role in the sequel. We can rephrase this result by saying that associate elements generate the same principal ideal.)
- Prove the converse of the previous result: if $a$ and $b$ generate the same principal ideal, then they are associates.
- Section 23, problem 16.
