(Not an expert!) Am a bit confused what you mean "modulo nr and ni" - modulo Gaussian integers? Then I think "they form a division ring" means the same as "The

(Not an expert!) Am a bit confused what you mean "modulo nr and ni" - modulo Gaussian integers? Then I think "they form a division ring" means the same as "The residue class ring modulo a Gaussian integer z is a field"? as in "The residue class ring modulo a Gaussian integer z is a field if and only if z is a Gaussian prime." https://en.wikipedia.org/wiki/Gaussian_integer#Residue_class_fields https://en.wikipedia.org/wiki/Gaussian_integer#Residue_class... There's a nice picture of the situation, just above that section.