On diagrams for abelian groups

AbstractLet ɒ be the ring of integers of an algebraic number field and p a prime ideal. Then if n is a positive integer, ɒ/pn is a primary ring with prime ideal p̄ = p/pn and the p̄i/p̄i+1 (0 ≤ i < n) are isomorphic groups under addition. Generalizing this idea, the author has defined the primary ring R with prime ideal N to be homogeneous if there is an integer k such that the additive groups RN, NN2, …, NkNk+1 are isomorphic and Nk+1 = 0. In determining the structure of the group of units of such a ring it became necessary to consider abelian groups with a certain type of series.The results in this paper are purely group-theoretic. We study groups which possess a particular type of series—this type of series we call a j-diagram. The structure of such groups is completely determined