# Flat modules and the behavior of a standard basis relative to an extension of the ground ring

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Let Âƒ : A â†’ B be a ring homomorphism and let J be an ideal of B. The purpose of this article is to examine the transfer of the properties of n-coherence and strong n-coherence (n â‰¥ 2) from a ring A to his amalgamated algebra A â‹ˆÂƒ J. Our results generate examples which...

- On noncommutative GrÃ¶bner bases over rings. Golod, E. S. // Journal of Mathematical Sciences;Jan2007, Vol. 140 Issue 2, p239
Let R be a commutative ring. It is proved that for verification of whether a set of elements { f Î±} of the free associative algebra over R is a GrÃ¶bner basis (with respect to some admissible monomial order) of the (bilateral) ideal that the elements f Î± generate it is sufficient to...

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A right ideal I of a ring R is small in case for every proper right ideal K of R, K + I â‰ R. A right R-module M is called PS-injective if every R-homomorphism f : aR â†’ M for every principally small right ideal aR can be extended to R â†’ M. A ring R is called right PS-injective...

- A Note on c-Separative Modules. Huanyin Chen // Kyungpook Mathematical Journal;2007, Vol. 47 Issue 3, p357
A right R-module P is c-separative provided that P âŠ• P â‰…c = P âŠ• Q &Arr; P â‰… Q for any right R-module Q. We get, in this paper, two sufficient conditions under which a right module is c-separative. A ring R is a hereditary ring provided that every ideal of R is...

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No abstract available.

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Strongly ?-supplemented and strongly cofinitely ?-supplemented modules are defined and some properties of strongly ?-supplemented and strongly cofinitely ?-supplemented modules are investigated. Let R be a ring. Then every R-module is strongly ?-supplemented if and only if R is perfect. The...

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It is proved that if R is an Abelian right pm-ring with a finite number of minimal prime ideals then R is a clean ring, which extends a main result in [1]. Some known results in [9] on clean elements of commutative reduced rings are extended to arbitrary Abelian rings. In this note, we also give...

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- Closure of Minimal Extensions. El Hajoui, M.; Miri, A. // Turkish Journal of Mathematics;2007, Vol. 31 Issue 4, p409
Let R be a commutative ring with a unit and M an R-module. In this paper we give a comparison between the F-closure in M of an R-submodule having a minimal extension and the closure of this minimal extension for the same Gabriel topology defined on the ring R. If J(R) âˆˆ F we prove that both...