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Note on strongly nil clean elements in rings

Aleksandra Kostić, Zoran Z. Petrović, Zoran S. Pucanović, Maja Roslavcev (2019)

Czechoslovak Mathematical Journal

Let R be an associative unital ring and let a R be a strongly nil clean element. We introduce a new idea for examining the properties of these elements. This approach allows us to generalize some results on nil clean and strongly nil clean rings. Also, using this technique many previous proofs can be significantly shortened. Some shorter proofs concerning nil clean elements in rings in general, and in matrix rings in particular, are presented, together with some generalizations of these results.

Notes on generalizations of Bézout rings

Haitham El Alaoui, Hakima Mouanis (2021)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we give new characterizations of the P - 2 -Bézout property of trivial ring extensions. Also, we investigate the transfer of this property to homomorphic images and to finite direct products. Our results generate original examples which enrich the current literature with new examples of non- 2 -Bézout P - 2 -Bézout rings and examples of non- P -Bézout P - 2 -Bézout rings.

On a notion of “Galois closure” for extensions of rings

Manjul Bhargava, Matthew Satriano (2014)

Journal of the European Mathematical Society

We introduce a notion of “Galois closure” for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of an S n degree n extension of fields. Moreover, we prove a number of properties of this construction; for example, we show that it is functorial and respects base change. We also investigate the behavior of this Galois closure construction for various natural classes of ring extensions.

On algebraic closures.

R. Raphael (1992)

Publicacions Matemàtiques

This is a description of some different approaches which have been taken to the problem of generalizing the algebraic closure of a field. Work surveyed is by Enoch and Hochster (commutative algebra), Raphael (categories and rings of quotients), Borho (the polynomial approach), and Carson (logic).Later work and applications are given.

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