Abstract:
This thesis establishes new results concerning the proof-theoretic strength
of two classic theorems of Ring Theory relating to factorization in integral
domains.
The first theorem asserts that if every irreducible is a prime, then every
element has at most one decomposition into irreducibles; the second states
that well-foundedness of divisibility implies the existence of an irreducible
factorization for each element.
After introductions to the Algebra framework used and Reverse Mathematics,
we show that the first theorem is provable in the base system of
Second Order Arithmetic RCA0, while the other is equivalent over RCA0 to
the system ACA0.
