Euclidean rings and prime factorization

Now we will talk about the factorization of elements in Euclidean rings. On pg.146 of “Topics in Algebra” by Herstein, it says: “Let be a Euclidean ring. Then every element in is either a unit in or can be written as the product of a finite number of prime elements in .” This seems elementary.Continue reading “Euclidean rings and prime factorization”

Euclidean rings and generators of ideals

This is to address a point that has just been glazed over in “Topics in Algebra” by Herstein. In a Euclidean ring, for any two elements such that . Also, there exists a function such that . We also know that the element with the lowest d-value generates the whole ring . The proof ofContinue reading “Euclidean rings and generators of ideals”