Definition:Equivalent Factorizations

Definition

Let $\struct {D, +, \circ}$ be an integral domain.

Let $x$ be a non-zero non-unit element of $D$.

Let there be two tidy factorizations of $x$:

$x = u_y \circ y_1 \circ y_2 \circ \cdots \circ y_m$

$x = u_z \circ z_1 \circ z_2 \circ \cdots \circ z_n$

These two factorizations are equivalent if one of the following equivalent statements holds:

$(1): \quad$ There exists a bijection $\pi: \set {1, \ldots, m} \to \set {1, \ldots, n}$ such that $y_i$ and $z_{\map \pi i}$ are associates of each other for each $i \in \set {1, \ldots, m}$.

$(2): \quad$ The multisets of principal ideals $\multiset {\ideal {y_i}: i = 1, \ldots, m}$ and $\multiset {\ideal {z_i}: i = 1, \ldots, n}$ are equal.

The equivalence of the definitions is shown by part $(3)$ of Principal Ideals in Integral Domain.

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