On graded almost-Bézout rings
Fatima-Zahra Guissi\,^1 and Najib Mahdou\,^2
\,^{1,2}Department of Mathematics, Faculty of Science and Technology, Fez, Morocco
Pages 1-10 | Received 14 May 2024, Accepted 28 August 2024, Published 10 July 2025
Abstract
Let R=\bigoplus_{\alpha\in\Gamma}R_{\alpha} be a commutative ring graded by an arbitrary abelian group \Gamma. We say that R is graded almost-Bézout ring if for each a,b\in h(R), there exists n\geq 1 and x\in h(R) such that (a^{n},b^{n})=(x). In this paper, we investigate the transfer of this property in different graded commutative ring extensions, namely, in graded trivial ring extensions (A \ltimes E) and graded amalgamated algebras along an ideal (A \bowtie^f J). Our aim is to provide examples of new classes of \Gamma-graded rings satisfying the above mentioned property.
Keywords: Graded almost Bézout ring, graded amalgamation of ring, gr-Bézout ring, graded trivial ring extension.
MSC numbers: 13A02, 13C13.
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