1 results
ON MODULES OVER COMMUTATIVE RINGS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 103 / Issue 3 / December 2017
- Published online by Cambridge University Press:
- 21 November 2017, pp. 341-356
- Print publication:
- December 2017
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- Article
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Our main purpose is to extend several results of interest that have been proved for modules over integral domains to modules over arbitrary commutative rings
$R$ with identity. The classical ring of quotients 
$Q$ of 
$R$ will play the role of the field of quotients when zero-divisors are present. After discussing torsion-freeness and divisibility (Sections 2–3), we study Matlis-cotorsion modules and their roles in two category equivalences (Sections 4–5). These equivalences are established via the same functors as in the domain case, but instead of injective direct sums 
$\oplus Q$ one has to take the full subcategory of 
$Q$-modules into consideration. Finally, we prove results on Matlis rings, i.e. on rings for which 
$Q$ has projective dimension 
$1$ (Theorem 6.4).
