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Gorenstein $n$-flat modules and their covers

C. Selvaraj, R. Udhayakumar and A. Umamaheswaran. Department of Mathematics Periyar University Salem - 636 011, TN, India.

AMS Subject classification (2000): 16D10, 16E30, 16D40.

In this paper, we introduce the notion of Gorenstein $n$-flat modules and Gorenstein $n$-absolutely pure modules. First, we prove that the direct limit of Gorenstein $n$-flat modules over a right $n$-coherent ring is again a Gorenstein $n$-flat module. Also we prove that over a right $n$-coherent ring, any pure submodule of a Gorenstein $n$-flat module is a Gorenstein $n$-flat module. Finally, the class of all Gorenstein $n$-flat left modules over a ring $R$ is a Kaplansky class and then we prove that all left modules over a right $n$-coherent ring have Gorenstein $n$-flat covers.

Keywords: $n$-coherent ring, Gorenstein flat module, Gorenstein $n$-absolutely pure module, Kaplansky class.

\section{Introduction}

Throughout this paper, $R$ denotes an associative ring with identity element. All modules, if not specified otherwise, are assumed to be left $R$-modules. We denote $R-Mod$ by the category of left $R$-modules. The character module $Hom_{\mathbb{Z}}(M,\mathbb{Q}/\mathbb{Z} )$ is denoted by $M^+$. We recall from \cite{Lee} a ring $R$ is right (resp. left) $n$-coherent (for integers $ n > 0$ or $n= \infty$) if every finitely generated submodule of a free right (resp. left) $R$-module whose projective dimension is $\leq n-1$ is finitely presented. Recall that a left (resp. right) $R$-module $M$ is called $n$-flat if $Tor^R_1(N,M)= 0 $ (resp. $Tor^R_1(M, N) = 0$) holds for all finitely presented right (resp. left) $R$-modules $N$ with projective dimension $ \leq n$. A left $R$-module $M$ is called $n$-absolutely pure if $Ext_R^1(N,M) = 0$ holds for all finitely presented left $R$-modules $N$ with projective dimension $\leq n$.

Enochs et al. \cite{Eno4} first introduced and studied Gorenstein flat modules over Gorenstein rings (that is, noetherian rings with finite self-injective dimension). One of the most interesting results is that over a right coherent ring the class of Gorenstein flat left modules and its right orthogonal class form a complete hereditary cotorsion pair, which is due to Enochs, Jenda and Lopez-Ramos \cite{Eno3}. Since the class of Gorenstein flat modules over a right coherent ring is closed under direct limits, one can get further that all left modules over a right coherent ring have Gorenstein flat covers. The existence of Gorenstein flat covers was first proved for modules over Gorenstein rings in \cite{Eno6}.

Recently, in \cite{Gan} Gang et al. proved all modules over a left $GF$-closed ring have Gorenstein flat covers. We proved all modules have $n$-flat covers in \cite{Sel}. These motivate us to introduce the notion of Gorenstein $n$-flat module and its cover. In this paper, we show that all left $R$-modules over right $n$-coherent ring have Gorenstein $n$-flat covers. This paper is organized as follows. In Section 3, we prove that over a right $n$-coherent ring, every direct limit of Gorenstein $n$-flat modules is again Gorenstein $n$-flat module. Also we introduce the notion of Gorenstein $n$-absolutely pure right $R$-modules and study the relation between them and we show that over a right $n$-coherent ring, any pure submodule of a Gorenstein $n$-flat module is Gorenstein $n$-flat. In Section 4, we prove given a ring $R$, the class of all Gorenstein $n$-flat left $R$-modules is a Kaplansky class and also we prove that all modules over a right $n$-coherent ring have Gorenstein $n$-flat covers and show that over a right $n$-coherent ring, Gorenstein $n$-flat cover of $M$ is $n$-flat cover of $M$.