Equivalence of Definitions of Associate in Integral Domain/Definition 1 Equivalent to Definition 2
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Theorem
The following definitions of the concept of Associate in the context of Integral Domain are equivalent:
Let $\struct {D, +, \circ}$ be an integral domain.
Let $x, y \in D$.
Definition 1
$x$ is an associate of $y$ (in $D$) if and only if they are both divisors of each other.
That is, $x$ and $y$ are associates (in $D$) if and only if $x \divides y$ and $y \divides x$.
Definition 2
$x$ and $y$ are associates (in $D$) if and only if:
- $\ideal x = \ideal y$
where $\ideal x$ and $\ideal y$ denote the ideals generated by $x$ and $y$ respectively.
Proof
We are to show that:
- $x \divides y \text{ and } y \divides x \iff \ideal x = \ideal y$
Thus:
\(\ds \) | \(\) | \(\ds x \divides y \text{ and } y \divides x\) | Definition 1 of Associate in Integral Domain | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \ideal y \subseteq \ideal x \text{ and } \ideal x \subseteq \ideal y\) | Element in Integral Domain is Divisor iff Principal Ideal is Superset | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \ideal x = \ideal y\) | Definition 2 of Set Equality |
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 62.3$ Factorization in an integral domain: $\text{(iii)}$