Marginal isomorphisms.
The classical right ring of quotients of a ring \(R\) is a superring where the regular elements (i.e. which are not zero divisors) of \(R\) are invertible, and all the elements of the superring are of the form \(xc^{-1}\) where \(x\) and \(c\) are elements of \(R\) and \(c\) is regular in \(R\). (Such a superring does not always exist.) Two modules \(G\) and \(H\) are margimorphic if there are maps \(a\colon G\to H\), \(b\colon H\to G\) such that the compositions \(ab\) and \(ba\) are regular (not zero divisors) in the respective endomorphism rings. A marginal summand is a direct summand of a margimorphic copy. The article develops a whole theory of marginal summands of finite direct sums of copies of a module \(G\) whose endomorphism ring has a classical right ring of quotients, and this quotient ring is semi-primary. The culmination of the theory is that margimorphism classes of marginal summands are in one-to-one correspondence with finitely generated projective modules over the quotient ring (Theorem 8.1). This correspondence preserves direct sums, leading to an Azumaya-Krull-Schmidt type decomposition theorem: marginal summands of \(G\) decompose uniquely to totally indecomposable modules (indecomposable modules up to margimorphism) up to margimorphism (Theorem 11.2). When the factor of the quotient ring with the Jacobson radical is a product of division rings, then the module uniquely decomposes into a finite direct sum of indecomposable modules (Corollary 13.7). The final section of the article contains classifications of modules whose endomorphism ring has a semi-primary classical right ring of quotients. A sample result is: the endomorphism ring has a semi-simple Artinian classical right ring of quotients if and only if the module is margimorphic to a finite direct sum of copies of some modules \(K_i\), such that \(\Hom(K_i,K_j)=0\) for \(i\neq j\), and the endomorphism ring of \(K_i\) has a classical right ring of quotients which is a division ring (Theorem 14.1).
scientific article; zbMATH DE number 5304626