Three frameworks for a general theory of factorization (Q1925576)

The authors discuss three different approaches for a general theory of factorization in integral domains: \(\tau\)-factorization, reduced \(\tau\)-factorization, and \(\Gamma\)-factorization. Let \(D\) be an integral domain, \(D^*\) the set of nonzero, nonunits of \(D\), and \(\tau\) a symmetric relation on \(D^*\). For \(a \in D^*, a = \lambda a_1 \cdots a_n\), where \(\lambda\) is a unit of \(D\), \(a_i \in D^*, n \geq 1\), and \(a_i \tau a_j\) for \(i \neq j\), is called a \(\tau\)-factorization of \(a\) and \(a_i\) is said to be a \(\tau\)-factor of \(a\). In the obvious way, one defines \(\tau\)-divides, \(\tau\)-atom, and \(\tau\)-prime. For example, if \(\tau = D^* \times D^*\), this gives the usual notions of factorization, and if \(a \tau b \Leftrightarrow (a,b) = D\), it gives the co-maximal theory of factorization of McAdam and Swan. The theory of reduced \(\tau\)-factorization is similar to that of \(\tau\)-factorization, but with \(\lambda = 1\). Finally, \(\Gamma\)-factorization is even more general and includes both previous factorization theories as special cases. These theories were introduced and studied by several students of the first author in their PhD Theses at the University of Iowa: A. M. Frazier (2006), S. M. Hamon (2007), R. M. Ortiz-Albino (2008), J. Reinkoester (2010), and J. Juett (2013).