Strongly J-n-Coherent rings

Abstract

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