We study varieties of rings with identity that satisfy an identity of the form
x
y
=
y
p
(
x
,
y
)
xy = yp(x,y)
, where every term of the polynomial
p
p
has degree greater than one. These varieties are interesting because they have definable principal congruences and are residually small. Let
V
\mathcal {V}
be such a variety. The subdirectly irreducible rings in
V
\mathcal {V}
are shown to be finite local rings and are completely described. This results in structure theorems for the rings in
V
\mathcal {V}
and new examples of noncommutative rings in varieties with definable principal congruences. A standard form for the defining identity is given and is used to show that
V
\mathcal {V}
also satisfies an identity of the form
x
y
=
q
(
x
,
y
)
x
xy = q(x,y)x
. Analogous results are shown to hold for varieties satisfying
x
y
=
q
(
x
,
y
)
x
xy = q(x,y)x
.