Abelian Groups That Are Direct Summands of Every Containing Abelian Group

It is a well known theorem that an abelian group G satisfying G = nG for every positive integer n is a direct summand of every abelian group H which contains G as a subgroup. It is the object of this note to generalize this theorem to abelian groups admitting a ring of operators, and to show that the corresponding conditions are not only sufficient but are at the same time necessary. Finally we show that every abelian group admitting a ring of operators may be imbedded in a group with the above mentioned properties; and it is possible to choose this "completion" of the given group in such a way that it is isomorphic to a subgroup of every other completion. Our investigation is concerned with abelian groups admitting a ring of operators. A ring R is an abelian group with regard to addition, its multiplication is associative, and the two operations are connected by the distributive laws. As the multiplication in R need not be commutative, we ought to distinguish left-, rightand two-sided ideals. Since, however, only left-ideals will occur in the future, we may use the term "ideals" without fear of confusion. Thus an ideal in R is a non-vacuous set M of elements in R with the property : If m', m" are elements in M, and if r', r " are elements in R, then r'm'±r"m" is an element in M. An abelian group G whose composition is written as addition is said to admit the elements in the ring R as operators (or shorter: G is an abelian group over R), if with every element r in R and g in G is connected their uniquely determined product rg so that this product is an element in G and so that this multiplication satisfies the associative and distributive laws. If G is an abelian group over R, then its subgroups M are characterized by the same property as the ideals MinR. We assume finally the existence of an element 1 in R so that g = lg for every g in G and r • 1 = 1 • r = r for every r in R. If x is any element in the abelian group G over Ry then its order N(x) consists of all the elements r in R which satisfy nc = 0. One verifies that every order N(x) is an ideal in 2?, and that N(x) =R if, and only if, x = 0. If M is an ideal in R, and if x is an element in G, then a subgroup