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The existence of relative pure injective envelopes

100%
Zareh-Khoshchehreh F., Divaani-Aazar K.
Colloquium Mathematicum
|
2013
|
tom 130
|
nr 2
251-264
EN
Let 𝓢 be a class of finitely presented R-modules such that R∈ 𝓢 and 𝓢 has a subset 𝓢* with the property that for any U∈ 𝓢 there is a U*∈ 𝓢* with U* ≅ U. We show that the class of 𝓢-pure injective R-modules is preenveloping. As an application, we deduce that the left global 𝓢-pure projective dimension of R is equal to its left global 𝓢-pure injective dimension. As our main result, we prove that, in fact, the class of 𝓢-pure injective R-modules is enveloping.
2
Content available remote

Cofiniteness of generalized local cohomology modules

100%
Divaani-Aazar K., Sazeedeh R.
Colloquium Mathematicum
|
2004
|
tom 99
|
nr 2
283-290
EN
Let 𝔞 denote an ideal of a commutative Noetherian ring R, and M and N two finitely generated R-modules with pd M < ∞. It is shown that if either 𝔞 is principal, or R is complete local and 𝔞 is a prime ideal with dim R/𝔞 = 1, then the generalized local cohomology module $H^i_{𝔞}(M,N)$ is 𝔞-cofinite for all i ≥ 0. This provides an affirmative answer to a question proposed in [13].
3
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A criterion for rings which are locally valuation rings

81%
Divaani-Aazar K., Esmkhani M., Tousi M.
Colloquium Mathematicum
|
2009
|
tom 116
|
nr 2
153-164
EN
Using the notion of cyclically pure injective modules, a characterization of rings which are locally valuation rings is established. As applications, new characterizations of Prüfer domains and pure semisimple rings are provided. Namely, we show that a domain R is Prüfer if and only if two of the three classes of pure injective, cyclically pure injective and RD-injective modules are equal. Also, we prove that a commutative ring R is pure semisimple if and only if every R-module is cyclically pure injective.
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