Let
be a ring with
identity,
be the
group of units of
and
be a positive
integer. We say that
is a
-unit if
. In particular,
if the ring
is for some
positive integer
we say that
is
a
-unit modulo
. We denote
by the set of
-units modulo
. We represent the
number of -units
modulo
by and the
ratio of
-units
modulo
by
,
where is
the Euler phi function. Recently, S. K. Chebolu proved that the solutions of the equation
are the divisors of
. Our main result finds
all positive integers
such that
for a given
.
Then we connect this equation with the Carmichael numbers and two of
their generalizations, namely, Knödel numbers and generalized Carmichael
numbers.
Keywords
diagonal property, diagonal unit, unit set of a ring,
$k$-unit, Carmichael number, Knödel number, Carmichael
generalized number
