Subrings of Artinian and Noetherian rings | Zendy

David Eisenbud | Zendy

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Subrings of Artinian and Noetherian rings

Author(s) -

David Eisenbud

Publication year - 1970

Publication title -

mathematische annalen

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 2.235

H-Index - 75

eISSN - 1432-1807

pISSN - 0025-5831

DOI - 10.1007/bf01350264

Subject(s) - mathematics , noetherian , artinian ring , pure mathematics , radical of a ring , semisimple module , noncommutative ring , algebra over a field , commutative ring , principal ideal ring , commutative property , ring (chemistry) , chemistry , organic chemistry

1. Main Results It is well known that if R C S are rings (rings in this paper have units but need not be commutative) such that S is finitely generated as a left R-module, then S is Noetherian or Artinian if R is. The converses of these statements are false (1; Example 1.0). In this paper we give a short homological proof that the converse statements do hold under slightly stronger hypotheses: Theorem 1. Let R C S be rings such that S is finitely generated as an R-module by elements which centralize R.

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