Let $\mathbf L_k$ be the holomorphic line bundle of degree $k \in \mathbb Z$ on projective line. Here, tuples $(k_1 k_2 k_3 k_4)$ for which there does not exists homogeneous non-split supermanifolds $CP^{1|4}_{k_1 k_4}$ associated with vector L_{−k_1} \oplus \mathbf L _{−k_2} L_{−k_3} L_{−k_4}$ are classified. \\For many types remaining tuples, listed cocycles that determine supermanifolds. \\P...

If R̂ is the pure-injective hull of a valuation ring R, it is proved that R̂ ⊗R M is the pure-injective hull of M , for every finitely generated Rmodule M . Moreover R̂ ⊗R M ∼= ⊕1≤k≤nR̂/AkR̂, where (Ak)1≤k≤n is the annihilator sequence of M . The pure-injective hulls of uniserial or polyserial modules are also investigated. Any two pure-composition series of a countably generated polyserial module a...

A module $$M$$ is said to be distributive (resp., uniserial) if the submodule lattice of a chain) Any uniserial but ring integers non-uniserial as $$\mathbb{Z}$$ -module. Direct sums (resp. modules are called semidistributive serial) modules. If $$A$$ with automorphism $$\varphi$$ , then we denote by $$A((x,\varphi))$$ skew Laurent series coefficient in which addition naturally defined and mult...

It is proved that a compact Kähler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kähler–Einstein manifolds. A stronger result is established for the case of Kähler surfaces. Without the compactness assumption, irreducible Kähler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situat...