论文标题
在最小环扩展
On minimal ring extensions
论文作者
论文摘要
令$ r $为具有身份的交换戒指。环$ r \ times r $可以通过对角映射$δ的扩展为$ r $的扩展名：r \ hookrightarrow r \ times r $，由$Δ（r）=（r）=（r，r）$用于r $。结果表明，对于任何$ a，b \ in r $，扩展$δ（r）[（a，b）] \ subset r \ times r $在且仅当理想$ <a-b> $是$ r $的最大理想时，才是最小环。还给出了$ r（+）r $的最大子环的完整分类。 von Neumann常规环$ r $的最小环扩展名是von Neumann常规环，或者是理想化$ r（+）r/\ mathfrak {m} $，其中$ \ mathfrak {m} m} \ in \ text {max {max}（max}（r）$。如果$ r \ subset t $是最小环的扩展名，而$ t $是一个积分域，则$（r：t）= 0 $ i时，并且仅当$ r $是一个字段，而$ t $是$ r $的最小字段扩展名，或$ r_j $是$ r_j $是高空的估价环，而$ t_ {j {j} $是$ t_ {j} $。
Let $R$ be a commutative ring with identity. The ring $R\times R$ can be viewed as an extension of $R$ via the diagonal map $Δ: R \hookrightarrow R\times R$, given by $Δ(r) = (r, r)$ for all $r\in R$. It is shown that, for any $a, b\in R$, the extension $Δ(R)[(a,b)] \subset R\times R$ is a minimal ring extension if and only if the ideal $<a-b>$ is a maximal ideal of $R$. A complete classification of maximal subrings of $R(+)R$ is also given. The minimal ring extension of a von Neumann regular ring $R$ is either a von Neumann regular ring or the idealization $R(+)R/\mathfrak{m}$ where $\mathfrak{m}\in \text{Max}(R)$. If $R\subset T$ is a minimal ring extension and $T$ is an integral domain, then $(R:T) = 0$ if and only if $R$ is a field and $T$ is a minimal field extension of $R$, or $R_J$ is a valuation ring of altitude one and $T_{J}$ is its quotient field.
