Such rings are called residually finite, or rings with the finite norm property. They have been studied, e.g. see the paper reviewed below.
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
$(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
$(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)