Abstract

A commutative non-associative division closed lattice-ordered ring with identity that is not an f -ring is presented. More conditions are provided to ensure that an associative division closed lattice-ordered ring is an f -ring. In particular, for a division closed lattice-ordered ring with identity, if it is Σ-clean or Σ-semiclean, then it is an f -ring. Finally it is shown that a ring with identity in which each partial order can be extended to a lattice order satisfying (x²ⁿ)⁻ = 0 for some integer n ≥ 1 must be an O*-ring.

中文翻译：

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除法闭ℓ-环和幂正L∗-环

摘要

提出了一个身份不是f环的可交换非结合除法闭合格序环。提供了更多条件以确保关联划分闭晶格有序环是f环。特别是，对于具有恒等式的除法闭晶格有序环，如果它是 Σ-clean 或 Σ-semiclean，那么它就是一个f环。最后表明，对于某个整数n ≥ 1 ，其中每个偏序都可以扩展到满足 ( x ^{2 n} ) ⁻ = 0的格序的具有身份的环必须是O * 环。