Below a special class of not necessarily associative or commutative rings
A
A
is considered which is characterized by the property that
A
A
has no nonzero nilpotent element and that a product of elements of
A
A
which is equal to zero remains equal to zero no matter how its factors are associated. It is shown that
(
A
,
⩽
)
(A, \leqslant )
is a partially ordered set where
x
⩽
y
x \leqslant y
if and only if
x
y
=
x
2
xy = {x^2}
. Also it is shown that
(
A
,
⩽
)
(A, \leqslant )
is infinitely distributive, i.e.,
r
sup
x
i
=
sup
r
x
i
r\sup {x_i} = \sup r{x_i}
. Finally, based on Zorn’s lemma it is shown that
A
A
is isomorphic to a subdirect product of not necessarily associative or commutative rings without zero divisors.