以下为句子列表：
英文: The present paper presents a commutativity theorem of rings with identity.
中文: 摘要给出有单位元的环的一个交换性定理。
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英文: Let {s~∨s~,y_0} be the union of two circles s~,s~ with common tangent point y_0.By defining a finite 'point sequence' to each closed path in {s~∨s~,y_0},a new proof was given to the non-commutativity of the fundamental group π_(s~∨s~,0).
中文: 设{s∨s,y0}为具有公共切点y0的两个圆s、s的并集,通过对s∨s中的每一个闭路径定义一个“点列”,给出基本群π(s∨s,y0)非交换性的一个新的证明.
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