0
$\begingroup$

Is there any set of mathematical objects that satisfies all of the following:

1) For each object $a$ in the set, $a^2$ is some multiples of $x$.

2) $ab$ is never multiples of $x$ where $a$ and $b$ are any different two objects in the set.

3) all objects commute - $ab = ba$ and $a+b = b+a$.

Does such set exists for all cardinality?

Edit: to specify, let objects be matrices.

  • 1
    What kind of structure? What is $x$? What is the motivation?2012-11-30
  • 0
    any structure - $x$ is a given one.2012-11-30
  • 0
    specified: matrices.2012-11-30
  • 0
    When you say "multiples of x", do you just mean scalar multiples ($2x, pi\cdot x$, etc.)?2012-12-01

2 Answers 2