Let be a three element set. Equip its power set
with two operations
and
defined as:
and
, for
. Then
is a ring. The zero-divisors in this ring are the singleton subsets and the two element subsets.
Recall that an element in a ring is a zero-divisor if
and there exists another
for which
(technically, a left zero-divisor).
The zero element in is obviously the empty set
. In addition, any singleton subset has an empty intersection with another singleton subset:
(we’ve assumed implicitly that
).
Consider a two element subset, e.g. . We have
. Similarly,
and
. So the two element subsets are also zero-divisors.
The entire set itself, namely is not a zero-divisor.
What is strange about these zero divisors as the title suggests? Nothing.
This concludes today’s snippet. Thanks for visiting!