A first important result in dimension theory is the fact that the Krull dimension of the ring K[X1, . . . , X`] is equal to ` when K is a field. In the literature this result is always obtained after some preliminary efforts that seem excessive for settling such an intuitive fact. For example, many authors rely on the principal ideal theorem of Krull, whose proof is very tricky. Matsumura [6,ch...

in this paper‎, ‎we give a complete proof of theorem 4.1(ii) and a new‎ ‎elementary proof of theorem 4.1(i) in [li and shen‎, ‎on the‎ ‎intersection of the normalizers of the derived subgroups of all‎ ‎subgroups of a finite group‎, ‎ j‎. ‎algebra, ‎323  (2010) 1349--1357]‎. ‎in addition‎, ‎we also give a generalization of baer's theorem‎.

Let (R,m) be a local ring (commutative and Noetherian). If R is complete (or, more generally, Henselian), one has the Krull-Schmidt uniqueness theorem for direct sums of indecomposable finitely generated R-modules. By passing to the m-adic completion b R, we can get a measure of how badly the Krull-Schmidt theorem can fail for a more general local ring. We assign to each finitely generated R-mo...

Suppose D is an integral domain with quotient eld K and that L is an extension eld of K. We show in Theorem 4 that if the complete integral closure of D is an intersection of Archimedean valuation domains on K, then the complete integral closure of D in L is an intersection of Archimedean valuation domains on L; this answers a question raised by All rings considered in this paper are assumed to...

— We prove a theorem of diophantine approximation between the field of formal power series in several variables and its completion for the Krull topology.

In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally, variants of the homological theorems are shown to hold in equal characteristic. This theory is then applied to Noetherian local rings in order to get: (i) over...

This paper focuses on the relationship between an $L$-subset and the system of level-elements induced by it, where the underlying lattice $L$ is a complete residuated lattice and the domain set of $L$-subset is an $L$-partially ordered set $(X,P)$. Firstly, we obtain the sufficient and necessary condition that an $L$-subset is represented by its system of level-elements. Then, a new representat...

We develop the theory of Krull dimension for S4-algebras and Heyting algebras. This leads to the concept of modal Krull dimension for topological spaces. We compare modal Krull dimension to other well-known dimension functions, and show that it can detect differences between topological spaces that Krull dimension is unable to detect. We prove that for a T1-space to have a finite modal Krull di...

We introduce and study in detail the notion of compatibility between valuations orderings real hyperfields. investigate their relation with induced on factor residue Much theory from fields can be generalized to hyperfields; we point out facts that cannot. generalize Baer-Krull theorem