Geodesic
Non-singular precovers over polynomial rings
Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 3, pp. 369-377 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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One of the results in my previous paper {\it On torsionfree classes which are not precover classes\/}, preprint, Corollary 3, states that for every hereditary torsion theory $\tau$ for the category $R$-mod with $\tau \geq\sigma$, $\sigma$ being Goldie's torsion theory, the class of all $\tau$-torsionfree modules forms a (pre)cover class if and only if $\tau$ is of finite type. The purpose of this note is to show that all members of the countable set $\frak M = \{R, R/\sigma (R), R[x_1,\dots ,x_n], R[x_1,\dots ,x_n]/\sigma(R[x_1,\dots ,x_n]), n \omega \}$ of rings have the property that the class of all non-singular left modules forms a (pre)cover class if and only if this holds for an arbitrary member of this set.
One of the results in my previous paper {\it On torsionfree classes which are not precover classes\/}, preprint, Corollary 3, states that for every hereditary torsion theory $\tau$ for the category $R$-mod with $\tau \geq\sigma$, $\sigma$ being Goldie's torsion theory, the class of all $\tau$-torsionfree modules forms a (pre)cover class if and only if $\tau$ is of finite type. The purpose of this note is to show that all members of the countable set $\frak M = \{R, R/\sigma (R), R[x_1,\dots ,x_n], R[x_1,\dots ,x_n]/\sigma(R[x_1,\dots ,x_n]), n \omega \}$ of rings have the property that the class of all non-singular left modules forms a (pre)cover class if and only if this holds for an arbitrary member of this set.
Classification : 16D50, 16D80, 16S90, 18E40
Keywords: hereditary torsion theory; torsion theory of finite type; Goldie's torsion theory; non-singular module; non-singular ring; precover class; cover class 
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Bican, Ladislav. Non-singular precovers over polynomial rings. Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 3, pp. 369-377. http://geodesic.mathdoc.fr/item/CMUC_2006_47_3_a0/