Setup:
One can study the ring of integers $\mathbb{Z}$ by looking at its quotient rings $\mathbb{Z}/k\mathbb{Z}$ often called "the Integers mod K". These quotient rings contain multiplicative subgroups that can sometimes have very small exponent,
A striking example is the multiplicative subgroup of the integers modulo 24, which we denote as $ \left( \mathbb{Z}/24\mathbb{Z} \right)^\times$. This group has only 8 elements contained in it, and very remarkably it has exponent 2. Meaning that for any element $a \in \left( \mathbb{Z}/24\mathbb{Z} \right)^\times, a^2 = 1$
Yet the original ring has 24 elements in it, which is a whole lot more than 2. By studying the Carmichael Function one can uncover other numerical gold where a VERY big ring contains a multiplicative subgroup with a very small exponent. (For example $ \left( \mathbb{Z}/240\mathbb{Z} \right)^\times $ where the original ring has 240 elements and yet the exponent here is only 4).
I'm interested in finding the bigger finite-commutative-rings whose multiplicative subgroups have very small exponents.
Question:
If we turn our attention to the gaussian integers $\mathbb{Z}[i]$ and look at quotient rings $R$. What is an example of the LARGEST finite quotient ring whose multiplicative subgroup has exponent 2?
Any quotient of $\Bbb{Z}[i]$ other than $\Bbb{Z}[i]/(1+i)$ and $\Bbb{Z}[i]/(2)$ contains the cyclic group $1,i,-1,-i$ so its multiplicative group won't be of exponent $2$.
To determinate the group structure of $\Bbb{Z}[i]/(3^m)^\times$ we go to the $p$-adic integers ($p=3$), starting with $\Bbb{Z}_p^\times=\langle \zeta_{p-1}\rangle (1+p)^{\Bbb{Z}_p}$ to obtain $\Bbb{Z}_3[i]^\times=\langle \zeta_8\rangle (1+3)^{\Bbb{Z}_3}(1+3i)^{\Bbb{Z}_3}$ and $$\Bbb{Z}[i]/(3^m)^\times=\Bbb{Z}_3[i]/(3^m)^\times=\langle \zeta_8\rangle (1+3)^{\Bbb{Z/3^{m-1}Z}}(1+3i)^{\Bbb{Z/3^{m-1}Z}}$$
The case $p\equiv 1\bmod 4$ is easier $$ \Bbb{Z}[i]/((2+i)^m)^\times=\Bbb{Z}_5/((2+i)^m)^\times=\Bbb{Z}_5/(5^m)^\times=\Bbb{Z}/(5^m)^\times= \langle \zeta_4\rangle (1+5)^{\Bbb{Z/5^{m-1}Z}}$$
From this and that Z[i] is a PID where p splits iff p≡3mod4 we know the group structure of $\Bbb{Z}[i]/(a)^\times$ for all $a$.