Let R be a commutative Noetherian local ring with residue class field k. In this paper, we mainly investigate direct summands of the syzygy modules of k. We prove that R is regular if and only if some syzygy module of k has a semidualizing summand. After that, we consider whether R is Gorenstein if and only if some syzygy module of k has a G-projective summand.

When a positive integer is expressed as a sum of squares, with each successive summand as large as possible, the summands decrease rapidly in size until the very end, where one may find two 4’s, or several 1’s. We find that the set of integers for which the summands are distinct does not have a natural density but that the counting function oscillates in a predictable way.

Let X be a complex K3 surface, $$\textrm{Diff}(X)$$ the group of diffeomorphisms and $$\textrm{Diff}_0(X)$$ identity component. We prove that fundamental contains free abelian countably infinite rank as direct summand. The summand is detected using families Seiberg–Witten invariants. moduli space Einstein metrics on used key ingredient in proof.

The aim of this paper is to investigate strong notion strongly ⨁-supplemented modules in module theory, namely ⨁-locally artinian supplemented modules. We call a M if it locally and its supplement submodules are direct summand. In study, we provide the basic properties particular, show that every summand supplemented. Moreover, prove ring R semiperfect with radical only projective R-module

We propose what be believe to be a novel approach to perform calculations for rational density functions using state space representations of the densities. By standard results from realisation theory, a rational probability density function is considered to be the transfer function of a linear system with generally complex entries. The stable part of this system is positive-real, which we call...