极大交换子环与Ore扩张的单性
Maximal commutative subrings and simplicity of Ore extensions
摘要：
本文的目的是描述Ore扩张环的单性的充要条件，重点是微分多项式环。证明了微分多项式环R[x;id，δ]是单的当且仅当它的中心是域且R是δ-单的.当R是交换的时，我们注意到R[x;σ，δ]中R的中心化子是包含R的极大交换子环，并且当σ=id时，我们证明了它与R[x;id，δ]的每个非零理想非平凡相交.利用这一点，我们证明了如果R是δ-单的且在R[x;id，δ]中是极大交换的，则R[x;id，δ]是单的.我们还表明，在某些条件下，在R的逆持有。
英文摘要：
The aim of this article is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x;id,δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x;σ,δ] is a maximal commutative subring containing R and, in the case when σ=id, we show that it intersects every non-zero ideal of R[x;id,δ] non-trivially. Using this we show that if R is δ-simple and maximal commutative in R[x;id,δ], then R[x;id,δ] is simple. We also show that under some conditions on R the converse holds.
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