The homological dimension of an abelian group as a module over its ring of endomorphisms

1965 ◽  
Vol 69 (4) ◽  
pp. 294-305 ◽  
Author(s):  
A. J. Douglas ◽  
H. K. Farahat
Keyword(s):  
Abelian Group ◽  
Homological Dimension ◽  
Ring Of Endomorphisms

Permutation endomorphisms and refinement of a theorem of Birkhoff

1960 ◽  
Vol 56 (4) ◽  
pp. 322-328 ◽  
Author(s):  
H. K. Farahat ◽  
L. Mirsky
Keyword(s):  
Abelian Group ◽  
Finite Number ◽  
Linear Combination ◽  
Finite Linear Combination ◽  
Usual Manner ◽  
Restricted Permutation ◽  
Image Position ◽  
Ring Of Endomorphisms ◽  
Finite Linear

Let be a free additive abelian group, and let be a basis of , so that every element of can be expressed in a unique way as a (finite) linear combination with integral coefficients of elements of . We shall be concerned with the ring of endomorphisms of , the sum and product of the endomorphisms φ, χ being defined, in the usual manner, by the equationsA permutation of a set will be called restricted if it moves only a finite number of elements. We call an endomorphism of a permutation endomorphism if it induces a restricted permutation of the basis .

scholarly journals Bounded topologies on endomorphism rings

2011 ◽  
Vol 20 (1) ◽  
pp. 1-3
Author(s):  
HOREA F. ABRUDAN ◽  
Keyword(s):  
Abelian Group ◽  
Endomorphism Rings ◽  
Ring Topology ◽  
Left And Right ◽  
Ring Of Endomorphisms

We prove in this note that the ring of endomorphisms of an infinite bounded Abelian group admits a nondiscrete right bounded ring topology. We give an example of an Abelian group whose ring of endomorphisms admits both nondiscrete left and right bounded topologies but does not admit a nondiscrete bounded ring topology.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

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