强n-凝聚环

设R是一个环,n是一个正整数.右R-模M称为强n-内射的,如果从任一自由右R-模F的任一n-生成子模到M的同态都可扩张为F到M的同态;右R-模V称为强n-平坦的,如果对于任一自由右R-模F的任一n-生成子模T,自然映射VT→VF是单的;环R称为左强n-凝聚的,如果自由左R-模的n-生成子模是有限表现的;环R称为左n-半遗传的,如果R的每个n-生成左理想是投射的.本文研究了强n-内射模,强n-平坦摸及左强n-凝聚环.通过模的强n-内射性和强n-平坦性概念,作者还给出了强n-凝聚环和n-半遗传环的一些刻画.

Strongly n-Coherent Rings

Let $R$ be a ring and $n$ a fixed positive integer. A right $R$-module $M$ is called strongly $n$-injective if every $R$-homomorphism from an $n$-generated submodule of a free right $R$-module $F$ to $M$ extends to a homomorphism of $F$ to $M$; a right $R$-module $V$ is said to be strongly $n$-flat, if for every $n$-generated submodule $T$ of a free right $R$-module $F$, the canonical map $V\otimes T\rightarrow V\otimes F$ is monic; a ring $R$ is called left strongly $n$-coherent if every $n$-generated submodule of a free left $R$-module is finitely presented; ring $R$ is said to be left $n$-semihereditary if every $n$-generated left ideal of $R$ is projective. The author studies strongly $n$-injective modules, strongly $n$-flat modules and left strongly $n$-coherent rings. Using the concepts of strongly $n$-injectivity and strongly $n$-flatness of modules, the author also presents some characterizations of strongly $n$-coherent rings and $n$-semihereditary rings.