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On a general theory of factorization in integral domains. (English) Zbl 1228.13001

The aim of this paper is to develop a general theory of factorization of elements in an arbitrary integral domain \(D\). This theory includes most of the previously studied cases of factorizations. More precisely, denote \(D^\# := D\setminus (U(D)\cup \{0\})\), where \(U(D)\) is the group of all invertible elements of \(D\). Let \(\,\tau\,\) be a relation on \(D^\#\), i.e., a subset of \(D^\#\times D^\#\). For \(a\in D^\#\) the authors define \(a= \lambda a_1\cdots a_n, \,\lambda \in U(D), \,a_i \in D^\#\), to be a \(\tau\)-factorization of \(a\) if \(a_i\tau a_j\) for each \(i\neq j\). One says that \(a\) is a \(\tau\)-product of the \(a_i\)’s and that \(a_i\) is a \(\tau\)-factor of \(a\). For \(a,b\in D^\#\), one says that \(a\) \(\tau\)-divides \(b\), written \(\,a|_\tau b\,\) if there exists \(\lambda \in U(D)\), \(n\geq 0\), \(i<n\), \(c_1,\ldots, c_n\in D^\#\), such that \(\lambda c_1 \cdots c_i a c_{i+1} \cdots c_n\) is a \(\tau\)-factorization of \(b\). One calls \(a= \lambda (\lambda^{-1} a)\) a trivial \(\tau\)-factorization of \(a\). The authors say that an element \(a\in D^\#\) is \(\tau\)-irreducible (resp. \(\tau\)-prime, resp. \(|_\tau\)-prime) if the only \(\tau\)-factorizations of \(a\) are the trivial ones (resp. if \(a|\lambda a_1\cdots a_n\), a \(\tau\)-factorization, then \(a|a_i\) for some \(i\), resp. if \(a|_\tau \lambda a_1\cdots a_n\), a \(\tau\)-factorization, then \(a|_\tau a_i\) for some \(i\)). The paper is illustrated with many relevant examples.

MSC:

13A05 Divisibility and factorizations in commutative rings
13G05 Integral domains
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
13E99 Chain conditions, finiteness conditions in commutative ring theory
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