$\Delta$-($\nabla$-)强Hopfian (Co-Hopfian)模与环

$\Delta$-($\nabla$-) Strongly Hopfian (Co-Hopfian) Modules and Rings

摘要: 作为强Hopfian模、强co-Hopfian模的推广，本文引入了$\Delta$ -强Hopfian模与$\nabla$ -强co-Hopfian模，并研究了其相关性质。实例表明，$\Delta$ -强Hopfian模未必是强Hopfian模，$\nabla$ -强co-Hopfian模也未必是强co-Hopfian模。此外，本文还针对局部投射、局部内射条件下的$\Delta$ -强Hopfian模与$\nabla$ -强co-Hopfian模作了细致研究。

Abstract: As generalizations of strongly Hopfian (co-Hopfian) modules, $\Delta$ -($\nabla$ -) strongly Hopfian (co Hopfian) modules are introduced and their properties are studied. Examples show that a $\Delta$ -strongly Hopfian module need not be a strongly Hopfian module and an $\nabla$ -strongly co-Hopfian module need not be a strongly co-Hopfian module. Moreover, $\Delta$ -strongly Hopfian and $\nabla$ -strongly co-Hopfian locally projective or locally injective modules are studied in detail.