Combining Local and Von Neumann Regular Rings
Combining Local and Von Neumann Regular Rings
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DOI：
10.1081/agb-120037405
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发表时间：
2004-12
影响因子：
0.7
通讯作者：
E. A. Osba;M. Henriksen;Osama Alkam
中科院分区：
文献类型：
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作者：
E. A. Osba;M. Henriksen;Osama Alkam
所有的环R都是交换的且有单位元。康特萨称R为VNL-环，如果a或1 − a有冯诺依曼逆，只要a ∈ R。示例结果：VNL-环的每个素理想都包含在唯一极大理想中。局部环和Von Neumann正则环是VNL，如果两个环的乘积是VNL，则两个环都是Von Neumann正则环，或者一个是Von Neumann正则环，另一个是VNL。整数模n的环<$n是VNL当且仅当（pq）2 <$n，只要p和q是不同的素数。环R上形式幂级数环R[[x]]是VNL当且仅当R是局部的. Tychonoff空间X上所有连续实值函数的环C（X）是VNL当且仅当X中至多有一点不是P点。所有已知的VNL-环都满足SVNL，即只要R的一个（有限）子集生成的理想是R的全部，它的一个成员就有一个冯诺依曼逆。我们证明了环R是SVNL当且仅当R的所有极大理想都是纯的，可能只有一个例外。证明了R（α）是SVNL当且仅当存在α0 ∈ I，使得R（α0）是SVNL，且对所有α ∈ I − {α0}，R（α）是VonNeumann正则环.是否每个VNL环都是SVNL是一个开放的问题。
Abstract All rings R considered are commutative and have an identity element. Contessa called R a VNL-ring if a or 1 − a has a Von Neumann inverse whenever a ∈ R. Sample results: Every prime ideal of a VNL-ring is contained in a unique maximal ideal. Local and Von Neumann regular rings are VNL and if the product of two rings is VNL, then both are Von Neumann regular, or one is Von Neumann regular and the other is VNL. The ring ℤ n of integers mod n is VNL iff (pq)2 ∤ n whenever p and q are distinct primes. The ring R[[x]] of formal power series over R is VNL iff R is local. The ring C(X) of all continuous real-valued functions on a Tychonoff space X is VNL if and only if at most one point of X fails to be a P-point. All known VNL-rings satisfy SVNL, namely whenever the ideal generated by a (finite) subset of R is all of R, one of its members has a Von Neumann inverse. We show that a ring R is SVNL if and only if all maximal ideals of R are pure except maybe one. We show that ∏α∈I R(α) is an SVNL if and only if there exists α0 ∈ I, such that R(α0) is an SVNL and for all α ∈ I − {α0}, R(α) is a Von Neumann regular ring. Whether every VNL-ring is an SVNL is an open question.