Finite conductor property in amalgamated algebra along an ideal

Recently, Xiang and Ouyang defined a (commutative unital) ring [Formula: see text] to be Nil[Formula: see text]-coherent if each finitely generated ideal of [Formula: see text] that is contained in Nil[Formula: see text] is a finitely presented [Formula: see text]-module. We define and study Nil[Formula: see text]-coherent modules and special Nil[Formula: see text]-coherent modules over any ring. These properties are characterized and their basic properties are established. Any coherent ring is a special Nil[Formula: see text]-coherent ring and any special Nil[Formula: see text]-coherent ring is a Nil[Formula: see text]-coherent ring, but neither of these statements has a valid converse. Any reduced ring is a special Nil[Formula: see text]-coherent ring (regardless of whether it is coherent). Several examples of Nil[Formula: see text]-coherent rings that are not special Nil[Formula: see text]-coherent rings are obtained as byproducts of our study of the transfer of the Nil[Formula: see text]-coherent and the special Nil[Formula: see text]-coherent properties to trivial ring extensions and amalgamated algebras.

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Commutative rings and modules that are Nil-coherent or special Nil-coherent

Ismaili

Dobbs

Mahdou

2017

J. Algebra Appl.

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On Steinitz-like properties in amalgamated algebras along an ideal

Maalmi

Mouanis

2019

J. Algebra Appl.

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Let [Formula: see text] be a ring homomorphism, [Formula: see text] be an ideal of [Formula: see text] and [Formula: see text] the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]. In this paper, we provide necessary and sufficient conditions for [Formula: see text] to be a Steinitz ring, semi-Steinitz ring, and weakly semi-Steinitz. Then we construct new original examples of weakly semi-Steinitz rings which are not semi-Steinitz rings and semi-Steinitz rings which are not Steinitz rings.

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Finite conductor property in amalgamated algebra along an ideal

Cited by 2 publications

References 8 publications

Commutative rings and modules that are Nil-coherent or special Nil-coherent

Commutative rings and modules that are Nil-coherent or special Nil-coherent

On Steinitz-like properties in amalgamated algebras along an ideal

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Finite conductor property in amalgamated algebra along an ideal

Cited by 2 publications

References 8 publications

Commutative rings and modules that are Nil*-coherent or special Nil*-coherent

Commutative rings and modules that are Nil*-coherent or special Nil*-coherent

On Steinitz-like properties in amalgamated algebras along an ideal

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Product

Resources

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Commutative rings and modules that are Nil-coherent or special Nil-coherent

Commutative rings and modules that are Nil-coherent or special Nil-coherent