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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 $I \ne \{0\}$

user26857
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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)

Bill Dubuque
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