A fundamental theorem of modular arithmetic (Q1705236)

In this paper the authors introduce the notion of irreducible, strongly irreducible and very strongly irreducible elements in a commutative ring \(R\) with identity which all coincide in case \(R\) is an integral domain. When \(D\) is a principal ideal domain and \(n\) is a nonzero nonunit of \(D\), then the authors give necessary and sufficient conditions on \(n\) for unique factorization of elements of the quotient ring \(D/\langle n \rangle\) into various type of irreducible elements. In the particular case when \(D=\mathbb{Z}\), interesting inferences are drawn. The proofs in the paper are lucid and self contained.