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Endomorphism rings generated using small numbers of elements
Author(s) -
Mesyan Zachary
Publication year - 2007
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdl038
Subject(s) - endomorphism , subring , mathematics , countable set , endomorphism ring , equivalence relation , diagonal , discrete mathematics , pure mathematics , equivalence (formal languages) , combinatorics , ring (chemistry) , geometry , chemistry , organic chemistry
Abstract Let R be a ring, M a nonzero left R ‐module, Ω an infinite set, and E = End R ( ⊕ Ω M ). Given two subrings S 1 , S 2 ⊆ E , write S 1 ≈ S 2 if there exists a finite subset U ⊆ E such that 〈 S 1 ∪ U 〉 = 〈 S 2 ∪ U 〉. We show that if M is simple and Ω is countable, then the subrings of E that are closed in the function topology and contain the diagonal subring of E (consisting of endomorphisms that take each copy of M to itself) fall into exactly two equivalence classes, with respect to the equivalence relation above. We also show that every countable subset of E is contained in a 2‐generator subsemigroup of E .