This paper aims at the study of the notions of periodic, UU and semiclean properties in various context of commutative rings such as trivial ring extensions, amalgamations and pullbacks. The results obtained provide new original classes of rings subject to various ring theoretic properties.

In this paper, we deal with the study of intermediate domains between a domain $R$ and a domain $T$ such that $T$ is an intersection of localizations of $R$, namely the pair $\left(R,T\right)$. More precisely, we study the pair $\left(R,R_d\right)$ and the pair $\left(R,\tilde{R}\right)$, where $R_d=\cap \left\{R_M\mid M\in \text{Max}\left(R\right),htM=dimR\right\}$ and $\tilde{R}=\cap \left\{R_M\mid M\in \text{Max}\left(R\right),htM\ge 2\right\}$. We prove that, if $R$ is a Jaffard domain, then $\left(R,R_d\left[n\right]\right)$ is a Jaffard pair, which generalize [5, Théorème 1.9]. We also show that if $R$ is an $S$-domain, then $\left(R,\tilde{R}\right)$ is a residually algebraic pair (that is for each intermediate domain $S$ between $R$ and $\tilde{R}$, if $Q$ is a prime ideal of $S$...

Let $R$ be a commutative ring with unity. The notion of maximal non valuation domain in an integral domain is introduced and characterized. A proper subring $R$ of an integral domain $S$ is called a maximal non valuation domain in $S$ if $R$ is not a valuation subring of $S$, and for any ring $T$ such that $R\subset T\subset S$, $T$ is a valuation subring of $S$. For a local domain $S$, the equivalence of an integrally closed maximal non VD in $S$ and a maximal non local subring of $S$ is established. The relation between $dim(R,S)$ and the number...

Let $R$ be a commutative ring with unity. The notion of maximal non $\lambda$-subrings is introduced and studied. A ring $R$ is called a maximal non $\lambda$-subring of a ring $T$ if $R\subset T$ is not a $\lambda$-extension, and for any ring $S$ such that $R\subset S\subseteq T$, $S\subseteq T$ is a $\lambda$-extension. We show that a maximal non $\lambda$-subring $R$ of a field has at most two maximal ideals, and exactly two if $R$ is integrally closed in the given field. A determination of when the classical $D+M$ construction is a maximal non $\lambda$-domain is given. A necessary condition is given...

A domain R is called a maximal non-Jaffard subring of a field L if R ⊂ L, R is not a Jaffard domain and each domain T such that R ⊂ T ⊆ L is Jaffard. We show that maximal non-Jaffard subrings R of a field L are the integrally closed pseudo-valuation domains satisfying dimv R = dim R + 1. Further characterizations are given. Maximal non-universally catenarian subrings of their quotient fields are also studied. It is proved that this class of domains coincides with the previous class when R is integrally...

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.