Can we characterize those Euclidean domains $D$ for which $D/I$ is finite for any ideal $I \ne \{0\}$ of $D$? Let $I$ be any ideal of $\mathbb Z[i]$ , then as $\mathbb Z[i]$ is euclidean domain , so $I=(z)$ for some gaussian integer $z$ ; so we can write every element of $\mathbb Z[i] / I$ as $x+(z)=qz+r+(z)=r+(z)$ where either $r=0$ or $N(r)<N(z)$ ; now for $r=0$ , every element of the form $x+(z)$ is nothing but $(z)$ and otherwise $N(r)$ is bounded above by $N(z)$ , so $N(r)$ , being a non-negative integer , can take only finitely many values , and since for a given $m \in \mathbb Z$ , $N(r)=N(r'+ir'')=r'^2+r''^2=m$ implies $r',r''$ can take only finitely many values , so $r$ can take only finitely many values ; hence the number of distinct elements of $\mathbb Z[i] / I$ is finite. Now my question is , can we , in general say  that for any euclidean domain $D$ and any ideal $I$ of it , $D/I$ is finite  ? If not , then can we characterize those Euclidean domains for which $D/I$ is finite for any ideal ? As far as I can see , I cannot carry the gaussian   domain approach in general , as there I used the finiteness of solution of $a^2+b^2=k$ in integers $a,b$ for given $k$ , to conclude $N(r)=k$ for a given $k$ has only finitely many solutions in $r$ in the euclidean  domain , It is true that if $N(r)=k$ for a given $k$ has only finitely many solutions in $r$ in the euclidean  domain $D$ , then $D/I$ is finite for any ideal $I$ , but I don't know whether this condition is necessary or not . Please help . Thanks in advance . 
$EDIT:$ In all of above $I \ne \{0\}$ 
 A: Such rings are called residually finite, or rings with the finite norm property FNP). Below is an entry point into the literature.

Levitz, Kathleen B.; Mott, Joe L.  $ $ Rings with finite norm property.
Canad. J. Math. 24 (1972), 557--565. 

Let  $A$  be a ring with  $A^2 \ne 0 ,$ and  $A^+$  the additive group
  of  $A$ . If each non-zero homomorphic image of  $A$  is finite, then 
  $A$  is said to be a ring with finite norm property (FNP ring). K. L.
  Chew and S. Lawn studied FNP rings with identity, which they called
  residually finite rings [same J. 22 (1970), 92--101; MR0260773 (41 #5396)]. In the paper under review, the authors extend the results of Chew and Lawn to arbitrary FNP rings. They also prove the following
  results: 
$(1)\ $ If $A$ is an FNP ring then $A^+$ is torsion and bounded, or
  torsion-free and reduced, or torsion-free and divisible. Henceforth,
  $A$ will be a commutative integral domain with $1$ and with quotient
  field $K$ .
$(2)\ $ Let L be a finite extension of $K$ ; if $A$ is an FNP ring,
  then so is  every intermediate ring of $L/A$ .
$(3)\ $ Let $A'$ be the integral closure of $A$ in $K$ ; then, $A$ is
  an FNP ring if and only if $A'$ is a Dedekind domain and $A_P$ is an
  FNP ring for every maximal ideal $P$ .
$(4)\ $ Let $K$ be of characteristic $0,$ then, every subring of $A$
  is an FNP ring  iff $K$ is a finite extension of the field of rational
  numbers.
$(5)\ $ Let $K \ne A$ be of prime characteristic; then, every subring
  of $A$ is an FNP ring iff $K$ is a finite extension of some $F(x),$
  where is the prime field of  $K$ and  $x$  is transcendental over $F$
  .

Review by H. Tominaga (AMS MR 45 #6872)
