The space between the two sides of the paper is an extra discrete ( noncontinuous ) , noncommutative space 介于这张纸两面之间的空间是一个额外的、离散的（不连续的）非可换空间。
In the past several years , noncommutative geometry has played more and more roles with the development of string theory 近几年，随着弦理论的发展，非对易几何才引起了人们的广泛关注。
Gopakumar et al succeeded in constructing the projection operator on noncommutative integral torus with generic r Gopakumar等人用另一种构造方法给出当uv = vu时不可对易torus上的投影算子。
Indeed , noncommutative geometry has arisen in at least three distinct but closely related contexts in string theory 实际上在弦理论中，非对易几何自然地出现在至少三种不同而又密切相关的背景里。
In the end , we study bare and pp rings of generalized power series on commutative and noncommutative rings 第四章分交换和非交换两种情况考虑了广义幂级数环上的bare性和pp性，主要是把[ 26 ，定理2
We disscuss the field theory in noncommutative space . the gauge invariance of noncommutative gauge field coupled with matter field is studied 我们讨论了非对易空间与规范场耦合的物质场的规范对称性。
In the third chapter , we study the annihilator ideals of power series and ore extensions on noncommutative rings and generalize [ 13 , proposition 3 . 1 ] 1 ] ，考虑ore扩张、幂级数环、斜幂级数环和迭代斜laurent多项式环上的情形，得出定理3
Noncommutative field theory is a quantum field theory on noncommutative space and means that fields are thoughts of as function over noncommutative space 非对易场论是建立在非对易空间上的量子场论，它意味着场量可以看作非对易空间的函数。
One would like to know to which extent properties of quantum field theories on commutative spaces also arise in quantum field theories on noncommutative space 我们总是想知道对易空间的量子场论的特点在多大程度上同样适用于非对易空间的量子场论。
In this paper , we mainly take two dimensional non - commutative chiral qcd2 model as an exameple to discuss the chiral anomaly in noncommutative field theory 在这些问题中，量子场论中的反常问题引起了人们的广泛兴趣。在这一方面，我们做了如下工作。
