Products over Finite Sets: a Single Factor, Vanishing in a Field, and Signs and Monotonicity in an Ordered Ring

In the setting of Commutative Rings, Fields and Ordered Fields: Standard Notation, let $A$ be a finite set.

Let $R$, with $+$, $\cdot$, $0$ and $1$, be a commutative ring, and let $f:A\to R$.

¶ If $x_{0}\in A$ and $f(x)=1$ for every $x\in A$ with $x\neq x_{0}$, then $\prod_{x\in A}f(x)=f(x_{0})$.

¶ If $R$ is a field, then $\prod_{x\in A}f(x)=0$ if and only if $f(x)=0$ for some $x\in A$.

Let now $R$, with a total order $\le$, be an ordered ring, and let $u,v:A\to R$.

¶ If $u(x)\ge0$ for every $x\in A$, then $\prod_{x\in A}u(x)\ge0$.

¶ If $0\le u(x)\le v(x)$ for every $x\in A$, then $\prod_{x\in A}u(x)\le\prod_{x\in A}v(x)$.