Division rings

A division ring is a mathematical structure similar to a field but with one key difference: while in a field every non-zero element has a multiplicative inverse (you can divide by it), in a division ring, this property holds for all non-zero elements, but multiplication might not be commutative (the order of multiplication matters). In simpler terms, you can divide by any number except zero, but multiplying two numbers may yield different results depending on the order you multiply them. Examples of division rings include the quaternions, which extend the concept of complex numbers.