Division Ring

A division ring is a mathematical structure similar to a field, where you can add, subtract, multiply, and divide elements (except by zero). The key difference is that in a division ring, multiplication may not be commutative—meaning the order of multiplication matters, so \(ab\) may not equal \(ba\). All non-zero elements in a division ring have inverses, allowing division. Think of it as an advanced number system where some properties are relaxed but core operations remain, making it useful in areas like algebra and geometry where non-commutative multiplication arises.