Real closures of commutative rings. I.
Real closures of commutative rings. II.
Regularity and intersections of bracket powers
Among reduced Noetherian prime characteristic commutative rings, we prove that a regular ring is precisely that where the finite intersection of ideals commutes with taking bracket powers. However, reducedness is essential for this equivalence. Connections are made with Ohm-Rush content theory, intersection-flatness of the Frobenius map, and various flatness criteria.
Relations between localizations and -adic completions in commutative domains
Relèvements de schémas et algèbres de Monsky-Washnitzer : théorèmes d équivalence et de pleine fidélité II
Remarks on commutative noetherian rings whose flat modules have flat injective envelopes
Remarks on Henselian rings.
Remarks on iteration of formal automorphisms.
Remarks on Noetherian local domains of dim 2.
Remarks on the Cayley-van der Waerden-Chow form.
Residue class rings of real-analytic and entire functions
Let 𝓐(ℝ) and 𝓔(ℝ) denote respectively the ring of analytic and real entire functions in one variable. It is shown that if 𝔪 is a maximal ideal of 𝓐(ℝ), then 𝓐(ℝ)/𝔪 is isomorphic either to the reals or a real closed field that is an η₁-set, while if 𝔪 is a maximal ideal of 𝓔(ℝ), then 𝓔(ℝ)/𝔪 is isomorphic to one of the latter two fields or to the field of complex numbers. Moreover, we study the residue class rings of prime ideals of these rings and their Krull dimensions. Use is made of...
Resolutions by mapping cones.
Résultats récents sur l'approximation des morphismes en algèbre commutative
Révêtements étales
Ring extensions with some finiteness conditions on the set of intermediate rings
A ring extension is said to be FO if it has only finitely many intermediate rings. is said to be FC if each chain of distinct intermediate rings in this extension is finite. We establish several necessary and sufficient conditions for the ring extension to be FO or FC together with several other finiteness conditions on the set of intermediate rings. As a corollary we show that each integrally closed ring extension with finite length chains of intermediate rings is necessarily a normal pair...
Rings generalized by tripotents and nilpotents
We present new characterizations of the rings for which every element is a sum of two tripotents and a nilpotent that commute. These extend the results of Z. L. Ying, M. T. Koşan, Y. Zhou (2016) and Y. Zhou (2018).
Rings with approximation property.
Root closure in commutative rings