Question

Let R be a principal ideal domain. For any two distinct nonzero elements f and g with no common irreducible factor, prove that ⟨f⟩ + ⟨g⟩ = ⟨1⟩.

Answer 1

To prove ➨f➩ + ➨g➩ = ➨1➩ in a principal ideal domain for elements f and g with no common factor, we use Bézout's Identity to find x, y in R such that xf + yg = 1, showing that the ideals' sum contains the ring's multiplicative identity.

Step-by-step explanation:

The subject of this question is principal ideal domain which falls under the category of abstract algebra in mathematics. To prove that for any two distinct nonzero elements f and g with no common irreducible factor in a principal ideal domain R, the sum of the ideals generated by f and g is equal to the entire ring, we use the fact that R is a principal ideal domain. Since f and g have no common irreducible factor, by Bézout's Identity, there exist elements x, y in R such that xf + yg = 1. The elements x and y can be thought of as multipliers that combine f and g to yield the multiplicative identity of the ring, which is 1. Thus, the sum of the principal ideals ➨f➩ and ➨g➩ must necessarily contain the multiplicative identity of the ring, making the sum of these ideals equal to the principal ideal generated by 1, denoted ➨1➩.