关于剩余类环的扩展的研究

作者对非结合环给出扩展的概念,即给定2个非结合环A和B,对任一非结合环R,称R是A被B的扩展,当且仅当A是R的理想且R/A≌B.对非结合环的扩展,文中证明了一个类似于Schreier群扩张定理的结果.作为应用,对给定的自然数m≥2,n≥2,文章刻画了模n的剩余类环Z_n被模m的剩余类环Z_m扩展所得到的有限环R的构造,证明了R可以用满足一定条件的自然数对(u,r)来描述,同时写出了R的理想和单侧理想的具体形状.作者还进一步证明,R是结合的当且仅当R=Z_nZ_m,且当R=Z_nZ_m时,R的每个理想都是Z_n的一个理想与Z_m的一个理想的直和,即此时R的理想是相对平凡的.

The Expansion of the Residue Class Ring

In this paper, the author introduces the concept of the expansion for non-associative rings. Let $A$ and $B$ be two non-associative rings, for an arbitrary non-associative ring $R$, we say that $R$ is the expansion of $A$ by $B$, if and only if $A$ is a two sided ideal of $R$ and $R/A\cong B$. First, for expansion, the author proves an analog to Schreier''s result on group extension. As an application, for fixed integers $m\geqslant2,\ n\geqslant2$, the author studies the construction of the finite ring $R$, where $R$ is the expansion of $Z_n$ (the residue class ring module $n$) by $Z_m$ (the residue class ring module $m$). It is shown that $R$ can be described by a certain pair $(u,r)\in\mathbb{N}\times\mathbb{N}$, and all the one-sided and two-sided ideals of $R$ are given out. Furthermore, it is proved that $R$ is associative if and only if $R=Z_n\oplus Z_m$, and once $R=Z_n\oplus Z_m$, then every ideal of $R$ is the direct sum of an ideal of $Z_n$ and an ideal of $Z_m$, hence ideals of $R$ are relatively trivial.