Vol. 15, No. 1, 1965

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Infinite sums in algebraic structures

Paul Katz and Ernst Gabor Straus

Vol. 15 (1965), No. 1, 181–190

The purpose of this note is an outline of an algebraic theory of summability in algebraic structures like abelian groups, ordered abelian groups, modules, and rings. “Infinite sums” of elements of these structures will be defined by means of homomorphisms satisfying some weak requirements of permanency which hold in all usual linear summability methods. It will turn out that several elementary well known theorems from the theory of infinite series, proved ordinarily by methods of analysis, (i.e. by use of some concept of a limit) are consequences of algebraic properties.

Mathematical Subject Classification
Primary: 40.50
Received: 21 August 1963
Revised: 11 February 1964
Published: 1 March 1965
Paul Katz
Ernst Gabor Straus