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Keywords:
hereditary torsion theory; torsion theory of finite type; Goldie’s torsion theory; non-singular module; non-singular ring; monoid ring; precover class; cover class
Summary:
Let $G$ be a multiplicative monoid. If $RG$ is a non-singular ring such that the class of all non-singular $RG$-modules is a cover class, then the class of all non-singular $R$-modules is a cover class. These two conditions are equivalent whenever $G$ is a well-ordered cancellative monoid such that for all elements $g,h\in G$ with $g < h$ there is $l\in G$ such that $lg = h$. For a totally ordered cancellative monoid the equalities $Z(RG) = Z(R)G$ and $\sigma (RG) = \sigma (R)G$ hold, $\sigma $ being Goldie’s torsion theory.
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