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 \(R\).