ring R

let $R$ be a ring with $a,b,c,d\in{R}$, $\bar{d}\subset{R}\land\bar{\mathbb{Z}}\subseteq{R}$, and operations $\oplus$ and $\otimes$,

$\mathbb{Z\equiv\bar{\mathbb{Z}}}$
$a\equiv{b}\equiv{c}$
$a\oplus{a}=a\oplus{a}\;,\;\forall{a,b}\in{R}$
$a\oplus{b}=b\oplus{a}\;,\;\forall{a,b}\in{R}$
$a\oplus{b\oplus{c}}=c\oplus(a\oplus{b})\;,\;\forall{a,b,c}\in{R}$
$a\otimes{a}=a\otimes{a}\;,\;a\in{R}$
$a\otimes{b}\neq{b}\otimes{a}\;,\;\forall{a,b}\in{R}\mid{a\neq{b}}$
$a\otimes(b\otimes{c})\neq(a\otimes{b})\oplus(a\otimes{c})\;,\;\forall{a,b,c}\in{R}\mid{a\neq{b}\neq{c}}$
$\exists{a}^{-1}\mida\otimes{a}^{-1}=1\;,\;\forall{a}\in{R}\mid{a,a^{-1}}\neq0$
$\bar{d}=\{d\in{\bar{\mathbb{Z}}}\mid{a}\otimes{d}=d\otimes{a}\;,\;\forall{a}\in{R}\}$
$\therefore\;a\stackrel{\scriptscriptstyle\mathrm{def}}{=}\left[\begin{eqnarray} a_{11}&&&a_{12}&&&\cdots&&&a_n \\ a_{21}&&&a_{22}&&&\vdots&&&a_{n2} \\ \vdots&&&\vdots&&&\ddots&&&\vdots \\ a_m&&&a_{m2}&&&\cdots&&&a_{\infty} \end{eqnarray}\right]$

let $\Box{ABCD}$ exist,

$\forall{\Box{ABCD}}\;(a)\mid!\exists({P_{\Box{ABCD}}}\land{A}_{\Box{ABCD}})}$
$P_{\Box{ABCD}}\stackrel{\scriptscriptstyle\mathrm{def}}{=}(4\otimes{a})\lor(a\otimes4)\;,\;(\bar{AB}\cong\bar{BC},\bar{CD},\bar{DA})\land(\bar{AB}=a)$
$A_{\Box{ABCD}}\stackrel{\scriptscriptstyle\mathrm{def}}{=}a\otimes{a}\;,\;(\bar{AB}\cong\bar{BC},\bar{CD},\bar{DA})\land(\bar{AB}=a)$

let $\Box{EFGH}$ exist,

$\forall\Box{EFGH}\;(a,b\mid\bar{EF}=a\mid\bar{FG}=b\mid{a\neq{b}})\mid!\exists{P_{\Box{EFGH}}}\land\exists{A}_{\Box{EFGH}}}$
$P_{\Box{EFGH}}\stackrel{\scriptscriptstyle\mathrm{def}}{=}[(2\otimes{a})\lor(a\otimes2)]\oplus[(2\otimes{b})\lor(b\otimes2)]\;,\;(\bar{EF}\cong\bar{GH})\land(\bar{FG}\cong\bar{HF})$
$A_{\Box{EFGH}}\stackrel{\scriptscriptstyle\mathrm{def}}{=}(a\otimes{b})\land(b\otimes{a})\;,\;(\bar{EF}\cong\bar{GH})\land(\bar{FG}\cong\bar{HF})$
$\displaystyle\frac{\delta}{\delta{a}}A_{\Box{EFGH}}\stackrel{\scriptscriptstyle\mathrm{def}}{=}a\otimes{b}$
$\therefore\;\displaystyle\frac{\delta}{\delta{a}}X\stackrel{\scriptscriptstyle\mathrm{def}}{=}a\otimes{b}$
$\displaystyle\frac{\delta}{\delta{b}}A_{\Box{EFGH}}\stackrel{\scriptscriptstyle\mathrm{def}}{=}b\otimes{a}$
$\therefore\;\displaystyle\frac{\delta}{\delta{b}}X\stackrel{\scriptscriptstyle\mathrm{def}}{=}b\otimes{a}$

this is interesting as, within ring $R$, at least squares and rectangles only have one possible perimeter, however rectangles have two simultaneous areas depending on how multiplication within the ring is carried out. it is therefore my conjecture that:

an equation within ring $R$ has $n!$ solutions, where $n$ represents the amount of variables being multiplied such that no variable is within the set $\bar{d}$ and such that no variable is equal to each other