> [!NOTE] Definition (Division Ring) > A division ring is a [[Rings|ring with unity]] $(R,+,\times)$ such that every non-[[Ring Zero Element|zero]] element is a [[Unit in a Ring|unit]]. **Equivalently**, we may assert that $R \setminus \{ 0 \} = R^{*},$ the [[Unit Group of Ring|unit group]] of $R.$ # Properties A division ring that is whose ring product is commutative is known as a [[Field (Algebra)|field]].