Some accessible treatment of algebraic
isomorphisms

Abstract. In this note, we unify many well-known theorems in abstract algebra as a result on a principal ideal domain (PID), Theorem 1.3. The proof entails an elucidating argument which establishes several isomorphism theorems in the theory of groups and fields at a stretch. This includes the use of the direct sum to interpret the method of constructing an extension ring of a given ring \(R\) as a quotient ring of the polynomial ring over \(R\) modulo a non-zero ideal, which in turn includes the case of the ring consisting of degree \(0\) polynomials as elements of the ring \(R\).

Key words and phrases. PID, homomorphism theorems, polynomial rings, direct sums, field theory.