Subdirect sums of rings

Abstract

Introduction.In much of the rather extensive theory of rings, the rings considered are restricted by certain finiteness assumptions.Under these assumptions the notion of direct sum plays an important role, but for general rings the usual theorems about direct sums are no longer valid.However, in certain important cases it turns out that the concept of subdirect sum is an appropriate generalization of that of direct sum.The purpose of this address is to summarize some of the more important known results concerning subdirect sums of rings.Following the introduction of the necessary definitions, we shall give an indication of the nature of these results.2. Notation and preliminary concepts.Let 5»-(i==l, 2, • • • , n) be given rings, and denote by S the set of all symbolsIf we define addition and multiplication of these symbols byandit is easy to verify that 5 is a ring which we call the direct sum of the rings Si (i = l, 2, • • • , n).The set of all elements of 5 of the formis clearly a subring S{ of 5 which is isomorphic to Si under the correspondenceFurthermore, S{ is closed under multiplication on either side by elements of 5 and hence is not only a subring of 5 but actually a twosided ideal in 5. If, in like manner, we define Si (i = 2, 3, • • • , n) y it is clear that every element of 5 is expressible uniquely in the form s{ +si + -••+ s: (si G 5/, * = 1, 2, • • • , »),

Locations

• Bulletin of the American Mathematical Society - View - PDF