On classical Krull dimension of group-graded rings

Abstract

For any ring R graded by a finite group, we give a bound on the classical Krull dimension of R in terms of the dimension of the initial component R e . It follows that if Re has finite classical Krull dimension, then the same is true of the whole ring R, too.Let G be a finite group with identity e. A ring R is said to be G-graded if R -© R g is a direct sum of additive subgroups R g and R g Rh Q R g h for all g, h € G. g€GThere are many results relating properties of a group-graded ring R = ® R g and its initial component R e , where e is the identity of the group (see [5,7,8] Rings with Krull dimension form an important class and have many nice properties (see [5]). Suppose that the set S = Spec(i?) of prime ideals of R satisfies a.c.c. Define the sets S a inductively. Let SQ be the set of all maximal elements in 5; and for each ordinal a denote by S a the set of all s £ S such that t € 5, t > s implies t E Sp for some /? < a. Then there exists the least ordinal a such that S a = S. This ordinal is called the classical Krull dimension of R. If it is finite, then it is also equal to the right Krull dimension of R defined on the lattice of right ideals of R (see [5, Chapter 6]).Denote by cl-K-dim(-R) the classical Krull dimension of R. For any ordinal a and positive integer n, we introduce ordinals a n , setting c*i = a + 1, a n+i = (a + l)(a n + 1)-We shall use the results on prime ideals due to Cohen and Montgomery [3] and prove the following theorem.

Partial Skew Group Rings Over Polycyclic by Finite Groups

“…For graded rings over finite groups A. V. Kelarev in [5] gave a bound to the classical Krull dimension in terms of the dimension of the initial component. One can easily see that any partial skew group ring is a group graded ring.…”

Incidence Semirings of Graphs and Visible Bases

“…We use standard terminology and refer readers to the monographs [5,8,9,14] and research articles [6,7,10,12] for more information. Throughout, the words 'graph' and 'digraph' mean a finite directed graph without multiple parallel edges but possibly with loops, and G = (V, E) is a graph with the set V of vertices and the set E of edges.…”

A Polynomial Ring Construction for the Classification of Data

“…2 Let S be a semigroup. An F-algebra R is said to be S-graded, if R = s∈S R s is a direct sum of F-modules R s and R s R t ⊆ R st , for all s, t ∈ S (see [14] and [13]). The Fmodules R s are called the homogeneous components of the grading.…”