## A strange zero divisor

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!