SOME PROPERTIES OF STAR OPERATIONS ON RING EXTENSIONS

Year 2021, Volume: 30 Issue: 30, 99 - 115, 17.07.2021

Lokendra Paudel Simplice Tchamna

https://doi.org/10.24330/ieja.969592

Cited By: 2

Abstract

Let $\star$ be a star operation on a ring extension $R\subseteq S$. A ring extension $R\subseteq S$ is called
Pr\"ufer $star$-multiplication extension (P$\star$ME) if $(R_{[\m]}, \m _{[\m]})$ is a Manis pair in $S$ for
every $\star$-maximal ideal $\m$ of $R$. We establish some results on star operations, and we study P$\star$ME
in pullback diagrams of type $\square$. We show that, for a
maximal ideal $\m$ of $R$, the extension $R_{[\m]} \subseteq S$ is
Manis if and only if $R[X]_{[\m R[X]]} \subseteq S[X]$ is a Manis
extension.

Keywords

Star operation, ring extension, Prüfer extension

References

• M. Fontana and M. Zafrullah, On $v$-domains: a survey. Commutative algebra–Noetherian and non-Noetherian perspectives, 145-179, Springer, New York, 2011.
• S. Gabelli and E. Houston, Ideal theory in pullbacks, In: Chapman S. T., Glaz S., eds. Non-Noetherian Commutative Ring Theory, Math. Appl., vol. 520, Kluwer Academic Publishers, Dordrecht (2000), 199-227.
• R. Gilmer, Multiplicative Ideal Theory, Corrected reprint of the 1972 edition, Queen’s Papers in Pure and Applied Mathematics, 90, Queen’s University, Kingston, ON, 1992.
• M. Griffin, Some results on $v$-multiplication rings, Canadian J. Math., 19 (1967), 710-722.
• E. G. Houston, S. B. Malik and J. Mott, Characterizations of $\star$-multiplication domains, Canadian Math. Bull., 27(1) (1984), 48-52.
• B. G. Kang, Prüfer $v$-multiplicative domains and the ring RrXsNv , J. Algebra, 123 (1989), 151-170.
• M. Knebusch and D. Zhang, Manis Valuations and Prüfer Extensions. I, Lecture Notes in Mathematics, 1791, Springer-Verlag, Berlin, 2002.
• M. Knebusch and T. Kaiser, Manis Valuations and Prüfer Extensions II, Lectures Notes in Mathematics, 2103, Springer, Cham, 2014.
• L. Paudel and S. Tchamna, Pullback diagrams and Kronecker function rings, Rocky Mountain J. Math., 49(7)(2019), 2267-2279.
• L. Paudel and S. Tchamna, A study of linked star operations, to appear in Bull. Korean Math. Soc.
• S. Tchamna, Multiplicative canonical ideals of ring extension, J. Algebra Appl., 16(4) (2017), 1750069 (15 pp).
• S. Tchamna, On ring extensions satisfying the star-hash property, Comm. Algebra, 48(5) (2020), 2081-2091.