A division algebra is a type of algebra in which every nonzero element has a multiplicative inverse. Other synonyms for this type of algebra include skew field, noncommutative field, and nonassociative field. Skew field is also a popular term commonly used in literature to refer to division algebra. Noncommutative field is used to describe a type of algebraic structure that doesn't commute across operations. This is a particularly important characteristic of a division algebra. Nonassociative field, on the other hand, refers to a type of algebra where the order of operations matter. In conclusion, division algebra, skew field, noncommutative field, and nonassociative field are all synonyms that refer to the same type of algebraic structure.