Ring of normal cones
A cone \(\gamma\) of a category \(\mathcal C\) is called a \textit{normal cone} if \(\gamma(A)\) is an isomorphism for some object \(A\). The authors define an \textit{RR}-category as an additive category \(\mathcal C\) with zero object which satisfies the following properties: 1) \(\mathcal C\) has subobjects; 2) every inclusion in \(\mathcal C\) splits; 3) the zero object is a subobject of each object; 4) the set of objects \(v\mathcal C\) with the partial order induced by the subobject relation is a relatively complemented lattice; 5) every \(f: A\to B\) in \(\mathcal C\) has a kernel \(j: K\to A\) such that \(j\) is an inclusion and every complement \(K'\) of \(K\) in \(A\) is a cokernel of \(j\); 6) every morphism in \(\mathcal C\) has an image and coimage satisfying certain properties; 7) for each \(A\in v\mathcal C\) there is a cone \(\gamma\) in \(\mathcal C\) with \(\gamma (A)=1_A\); 8) every subset of \(v\mathcal C\) which has an upper bound in \(v\mathcal C\) contains a maximal element. It is proved that if \(\mathcal C\) is an \textit{RR}-category then the set of all normal cones forms a regular ring. If \(X\) is an infinite-dimensional vector space, then the category \(\mathcal D\) with \(v\mathcal D=\{M: M\) is a finite-dimensional subspace of \(X\}\) and linear transformations is an example of an \textit{RR}-category; the set of normal cones in \(\mathcal D\) coincides with the set of all finite-rank operators on \(X\).
scientific article; zbMATH DE number 5924949