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Mappings of degree 5, part I

M. Maciejewski, A. Prószyński (2009)

Colloquium Mathematicae

The class of linear (resp. quadratic) mappings over a commutative ring is determined by a set of equation-type relations. For the class of homogeneous polynomial mappings of degree m ≥ 3 it is so over a field, and over a ring there exists a smallest equationally definable class of mappings containing the preceding one. It is proved that generating relations determining that class can be chosen to be strong relations (that is, of the same form over all commutative rings) if{f} m ≤ 5. These relations...

Multiplication modules and related results

Shahabaddin Ebrahimi Atani (2004)

Archivum Mathematicum

Let R be a commutative ring with non-zero identity. Various properties of multiplication modules are considered. We generalize Ohm’s properties for submodules of a finitely generated faithful multiplication R -module (see [8], [12] and [3]).

On commutative rings whose maximal ideals are idempotent

Farid Kourki, Rachid Tribak (2019)

Commentationes Mathematicae Universitatis Carolinae

We prove that for a commutative ring R , every noetherian (artinian) R -module is quasi-injective if and only if every noetherian (artinian) R -module is quasi-projective if and only if the class of noetherian (artinian) R -modules is socle-fine if and only if the class of noetherian (artinian) R -modules is radical-fine if and only if every maximal ideal of R is idempotent.

On finitely generated multiplication modules

R. Nekooei (2005)

Czechoslovak Mathematical Journal

We shall prove that if M is a finitely generated multiplication module and A n n ( M ) is a finitely generated ideal of R , then there exists a distributive lattice M ¯ such that S p e c ( M ) with Zariski topology is homeomorphic to S p e c ( M ¯ ) to Stone topology. Finally we shall give a characterization of finitely generated multiplication R -modules M such that A n n ( M ) is a finitely generated ideal of R .

On n-derivations and Relations between Elements rⁿ-r for Some n

Maciej Maciejewski, Andrzej Prószyński (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

We find complete sets of generating relations between the elements [r] = rⁿ - r for n = 2 l and for n = 3. One of these relations is the n-derivation property [rs] = rⁿ[s] + s[r], r,s ∈ R.

On prime modules over pullback rings

Shahabaddin Ebrahimi Atani (2004)

Czechoslovak Mathematical Journal

First, we give a complete description of the indecomposable prime modules over a Dedekind domain. Second, if R is the pullback, in the sense of [9], of two local Dedekind domains then we classify indecomposable prime R -modules and establish a connection between the prime modules and the pure-injective modules (also representable modules) over such rings.

On prime submodules and primary decomposition

Yücel Tiraş, Harmanci, Abdullah (2000)

Czechoslovak Mathematical Journal

We characterize prime submodules of R × R for a principal ideal domain R and investigate the primary decomposition of any submodule into primary submodules of R × R .

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