SOME PROPERTIES OF STAR OPERATIONS ON RING EXTENSIONS

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

1. M. Fontana and M. Zafrullah, On $v$-domains: a survey. Commutative algebra–Noetherian and non-Noetherian perspectives, 145-179, Springer, New York, 2011.
2. 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.
3. 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.
4. M. Griffin, Some results on $v$-multiplication rings, Canadian J. Math., 19 (1967), 710-722.
5. E. G. Houston, S. B. Malik and J. Mott, Characterizations of $\star$-multiplication domains, Canadian Math. Bull., 27(1) (1984), 48-52.
6. B. G. Kang, Prüfer $v$-multiplicative domains and the ring RrXsNv , J. Algebra, 123 (1989), 151-170.
7. M. Knebusch and D. Zhang, Manis Valuations and Prüfer Extensions. I, Lecture Notes in Mathematics, 1791, Springer-Verlag, Berlin, 2002.
8. M. Knebusch and T. Kaiser, Manis Valuations and Prüfer Extensions II, Lectures Notes in Mathematics, 2103, Springer, Cham, 2014.

9. L. Paudel and S. Tchamna, Pullback diagrams and Kronecker function rings, Rocky Mountain J. Math., 49(7)(2019), 2267-2279.
10. L. Paudel and S. Tchamna, A study of linked star operations, to appear in Bull. Korean Math. Soc.
11. S. Tchamna, Multiplicative canonical ideals of ring extension, J. Algebra Appl., 16(4) (2017), 1750069 (15 pp).
12. S. Tchamna, On ring extensions satisfying the star-hash property, Comm. Algebra, 48(5) (2020), 2081-2091.