noncommutative free ring

例句：

1. 1. The noncommutative property of the material is a significantfactor in its performance.
   这种材料的非交换性质是其性能的一个重要因素。
2. 2. In mathematics, the noncommutativering is a type of ring that doesnot satisfy the commutative property.
   在数学中，非交换环是一种不满足交换性质的环。
3. 3. The noncommutative algebraic structure has beenstudied extensively in the field of quantum mechanics.
   量子力学领域对非交换代数结构进行了广泛的研究。