{"title":"通过无穷级数的幂等性看群和环的心性","authors":"Abolfazl Tarizadeh","doi":"arxiv-2409.02488","DOIUrl":null,"url":null,"abstract":"An important classical result in ZFC asserts that every infinite cardinal\nnumber is idempotent. Using this fact, we obtain several algebraic results in\nthis article. The first result asserts that an infinite Abelian group has a\nproper subgroup with the same cardinality if and only if it is not a Pr\\\"ufer\ngroup. In the second result, the cardinality of any monoid-ring $R[M]$ (not\nnecessarily commutative) is calculated. In particular, the cardinality of every\npolynomial ring with any number of variables (possibly infinite) is easily\ncomputed. Next, it is shown that every commutative ring and its total ring of\nfractions have the same cardinality. This set-theoretic observation leads us to\na notion in ring theory that we call a balanced ring (i.e. a ring that is\ncanonically isomorphic to its total ring of fractions). Every zero-dimensional\nring is a balanced ring. Then we show that a Noetherian ring is a balanced ring\nif and only if its localization at every maximal ideal has zero depth. It is\nalso proved that every self-injective ring (injective as a module over itself)\nis a balanced ring.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"75 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Cardinality of groups and rings via the idempotency of infinite cardinals\",\"authors\":\"Abolfazl Tarizadeh\",\"doi\":\"arxiv-2409.02488\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"An important classical result in ZFC asserts that every infinite cardinal\\nnumber is idempotent. Using this fact, we obtain several algebraic results in\\nthis article. The first result asserts that an infinite Abelian group has a\\nproper subgroup with the same cardinality if and only if it is not a Pr\\\\\\\"ufer\\ngroup. In the second result, the cardinality of any monoid-ring $R[M]$ (not\\nnecessarily commutative) is calculated. In particular, the cardinality of every\\npolynomial ring with any number of variables (possibly infinite) is easily\\ncomputed. Next, it is shown that every commutative ring and its total ring of\\nfractions have the same cardinality. This set-theoretic observation leads us to\\na notion in ring theory that we call a balanced ring (i.e. a ring that is\\ncanonically isomorphic to its total ring of fractions). Every zero-dimensional\\nring is a balanced ring. Then we show that a Noetherian ring is a balanced ring\\nif and only if its localization at every maximal ideal has zero depth. It is\\nalso proved that every self-injective ring (injective as a module over itself)\\nis a balanced ring.\",\"PeriodicalId\":501136,\"journal\":{\"name\":\"arXiv - MATH - Rings and Algebras\",\"volume\":\"75 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Rings and Algebras\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.02488\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.02488","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Cardinality of groups and rings via the idempotency of infinite cardinals
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 Pr\"ufer
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.