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.
