通过无限基数的幂等性研究群与环的基数
Cardinality of groups and rings via the idempotency of infinite cardinals
AI总结：
本文利用 ZFC 中无限基数幂等性这一集合论事实，研究群与环的基数性质，证明了无限阿贝尔群具有同基数真子群的充要条件，计算了幺半群环与多项式环的基数，并引入平衡环概念，刻画了诺特环与自内射环成为平衡环的条件。
摘要：
ZFC 中一个重要的经典结果断言：每一个无限基数都是幂等的。利用这一事实，我们在本文中得到了若干代数结果。第一个结果断言：一个无限阿贝尔群具有与其基数相同的真子群，当且仅当它不是 Prüfer 群。在第二个结果中，计算了任意幺半群环 $R[M]$（不必交换）的基数。特别地，任意多个变元（可能无限）的多项式环的基数都可以容易地算出。接下来证明了每个交换环与其全分式环具有相同的基数。这一集合论观察引导我们提出环论中的一个概念，我们称之为平衡环（即典范同构于其全分式环的环）。每个零维环都是平衡环。然后我们证明：一个诺特环是平衡环，当且仅当它在每个极大理想处的局部化具有零深度。还证明了每个自内射环（作为其自身上的模是内射的）都是平衡环。
英文摘要：
An important classical result in ZFC asserts that every infinite cardinal number is idempotent. Using this fact, we obtain several algebraic results in this article. The first result asserts that an infinite Abelian group has a proper subgroup with the same cardinality if and only if it is not a Prüfer group. In the second result, the cardinality of any monoid-ring $R[M]$ (not necessarily commutative) is calculated. In particular, the cardinality of every polynomial ring with any number of variables (possibly infinite) is easily computed. Next, it is shown that every commutative ring and its total ring of fractions have the same cardinality. This set-theoretic observation leads us to a notion in ring theory that we call a balanced ring (i.e. a ring that is canonically isomorphic to its total ring of fractions). Every zero-dimensional ring is a balanced ring. Then we show that a Noetherian ring is a balanced ring if and only if its localization at every maximal ideal has zero depth. It is also proved that every self-injective ring (injective as a module over itself) is a balanced ring.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
