标题： 具有半局部自同态环之间的模同态 显示英文标题

标题： Homomorphisms with semilocal endomorphism rings between modules

摘要： 我们研究了范畴$\operatorname{Morph}(\operatorname{Mod} R)$，其对象是两个右$R$-模之间的所有态射。 在$\operatorname{Morph}(\operatorname{Mod} R)$中，对象的自环为半局部的$\operatorname{Morph}(\operatorname{Mod} R)$的行为与自环为半局部的模的行为非常相似。 例如，直和分解 $\oplus_{i=1}^nM_i$（即分块对角分解），其中 $\operatorname{Morph}(\operatorname{Mod} R)$ 的每个对象 $M_i$ 表示一个态射 $\mu_{M_i}\colon M_{0,i}\to M_{1,i}$，并且所有模 $M_{j,i}$ 都具有局部自同态环 $\operatorname{End}(M_{j,i})$，这种分解依赖于两个不变量。 这种行为与有限 Goldie 维度的串型模的直和分解非常相似，后者也依赖于两个不变量（单生成类和上生成类）。 当所有的模$M_{j,i}$都是单链模时，直接和的直接和分解（块对角分解）$\oplus_{i=1}^nM_i$取决于四个不变量。 显示英文摘要

摘要： We study the category $\operatorname{Morph}(\operatorname{Mod} R)$ whose objects are all morphisms between two right $R$-modules. The behavior of objects of $\operatorname{Morph}(\operatorname{Mod} R)$ whose endomorphism ring in $\operatorname{Morph}(\operatorname{Mod} R)$ is semilocal is very similar to the behavior of modules with a semilocal endomorphism ring. For instance, direct-sum decompositions of a direct sum $\oplus_{i=1}^nM_i$, that is, block-diagonal decompositions, where each object $M_i$ of $\operatorname{Morph}(\operatorname{Mod} R)$ denotes a morphism $\mu_{M_i}\colon M_{0,i}\to M_{1,i}$ and where all the modules $M_{j,i}$ have a local endomorphism ring $\operatorname{End}(M_{j,i})$, depend on two invariants. This behavior is very similar to that of direct-sum decompositions of serial modules of finite Goldie dimension, which also depend on two invariants (monogeny class and epigeny class). When all the modules $M_{j,i}$ are uniserial modules, the direct-sum decompositions (block-diagonal decompositions) of a direct-sum $\oplus_{i=1}^nM_i$ depend on four invariants.