A variety
V
\mathcal {V}
of rings has definable principal congruences (DPC) if there is a first order sentence defining principal two-sided ideals for all rings in
V
\mathcal {V}
. The key result is that for any ring
R
R
,
V
(
M
n
(
R
)
)
V({M_n}(R))
does not have DPC if
n
⩾
2
n \geqslant 2
. This allows us to show that if
V
(
R
)
V(R)
has DPC, then
R
R
is a polynomial identity ring. Results from the theory of PI rings are used to prove that for a semiprime ring
R
R
,
V
(
R
)
V(R)
has DPC if and only if
R
R
is commutative. An example of a finite, local, noncommutative ring
R
R
with
V
(
R
)
V(R)
having DPC is given.