Definition:Trivial Factorization
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Definition
Let $\struct {D, +, \circ}$ be an integral domain.
Let $\struct {U_D, \circ}$ be the group of units of $\struct {D, +, \circ}$.
A factorization in $\struct {D, +, \circ}$ of the form $x = u \circ y$, where $u \in U_D$ (that is, where $x$ is an associate of $y$) is called a trivial factorization.
Non-Trivial Factorization
A factorization in $\struct {D, +, \circ}$ of the form $x = z \circ y$, where neither $y$ nor $z$ is a unit of $D$, is called a non-trivial factorization.
Linguistic Note
The spelling factorization is the US English version.
The UK English spelling is factorisation, but the tendency is for the literature to use the factorization form.
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 62$. Factorization in an integral domain