单环为除环的一个充分条件及交换单环的分类

运用环论中理想的性质，得到单环为除环的一个充分条件及其推论。同时运用群论中Lagrange定理的较弱逆命题，推出没有真子群的群必为素数阶有限群的结论；进一步综合所得结果，得到交换单环的一个分类。即交换单环要么是域要么是基数为素数的零乘环。

A Sufficient Condition of A Simple Ring to Be A Division Ring and the Classification of the Commutative Simple Ring

Using the properties of the ideal of the ring theory,we obtain a sufficient condition of a simple ring to be a division ring and its corollary,and with the weaker inverse proposition of Lagrange's theorem in the group theory,we get the conclusion that the group without proper subgroup must be a finite group of prime order.Furthermore,based on the results,we get a classification of the commutative simple ring,that is the commutative simple ring either a field or a zero multiplication ring with the prime base number.