## On a certain determinant for a U.F.D.

### Volume 171 / 2023

#### Abstract

Let $R$ be a unique factorization domain (U.F.D.). Let $P=\{p_i\}$ be a prime system of $R$, that is, $P$ is a well-ordered set of non-associate prime elements of $R$ and $P$ is complete. An element $x\in R$ is called a $P$-number if $x$ is a product of elements from the prime system $P$. The set consisting of $1_R$ and all $P$-numbers is denoted by $\overline {P}$. We call an element $d\in R$ a $P$-divisor of $x$ in $R$ if $d\,|\,x$ and $d\in \overline {P}$. We use $(x,y)_P$ to denote the greatest common $P$-divisor of non-zero elements $x$ and $y$ in $R$. Let $W=\{w_1,\ldots ,w_n\}$ be a set of non-associate, non-zero elements of $R$ and $e\ge 1$ an integer. The set $W$ is called $P$-factor-closed if all the elements of $W$ belong to $\overline {P}$ and $d$ is an associate of some element $w’\in W$ whenever $d$ divides an element $w\in W$. Let $((w_i,w_j)^e_P))$ denote the $n\times n$ power GCD $P$-matrix on $W$ whose $(i,j)$-entry is the $e$th power of $(w_i,w_j)_P$. We give the formula for the determinant of the $(n-1)\times (n-1)$ power GCD $P$-matrix $((W\smallsetminus \{w_t\})^e)=((w_i,w_j)^e_P)_{\substack {1\le i, j\le n\\ i\neq t,\, j\neq t}}$, where $W$ is $P$-factor-closed and $1\le t\le n$. Our result extends Smith’s famous determinant and the Lin–Hong theorem. It also generalizes the Beslin–Kassar theorem and the Hong–Zhou–Zhao theorem.