关于剩余类环的扩展的研究
The Expansion of the Residue Class Ring

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$R/A\cong B$. 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$m\geqslant2$, $n\geqslant2$, 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$R=Z_{n}\oplus Z_{m}$, 且当 $R=Z_{n}\oplus Z_{m}$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.
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.