Ladislav Bican
Non-singular precovers over polynomial rings

Comment.Math.Univ.Carolin. 47,3 (2006) 369-377.

Abstract:One of the results in my previous paper {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 $\frak M = \{R, R/\sigma (R), R[x_1,...,x_n], R[x_1,...,x_n]/\sigma (R[x_1,...,x_n]), n <\omega \}$ 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.

Keywords: hereditary torsion theory, torsion theory of finite type, Goldie's torsion theory, non-singular module, non-singular ring, precover class, cover class
AMS Subject Classification: 16S90, 18E40, 16D80