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In the paper, a Frattini-like subgroup associated with a class ?? of simple groups is introduced and analyzed. The corresponding ??-saturated formations are exactly the ??-local ones introduced by Förster. Our techniques are also useful to highlight the properties and behavior of ?-local formations. In fact, extensions and improvements of several results of Shemetkov are natural consequences of our study. [ABSTRACT FROM AUTHOR]

This paper gives an explicit formula for the Hilbert series of algebras of Veronese type. [ABSTRACT FROM AUTHOR]

A right ideal I of a ring R is called a comparizer right ideal if for all right ideals A, B of R, either A ⊆ B or BI ⊆ A. For every ring R there exists the largest comparizer ideal C1(R) of R, and higher comparizer ideals Cα(R) can be defined inductively. In this paper, comparizer right ideals and relationships between the iterated comparizer ideal and some classical radicals of R are studied. Obtained results generalize some properties of right chain rings, right distributive rings, right Bezout rings and rings with comparability. [ABSTRACT FROM AUTHOR]

In this paper, we discuss generalizations of the concepts of good filtration dimension and Weyl filtration dimension, introduced by Friedlander and Parshall for algebraic groups, to properly stratified algebras. We introduce the notion of the finitistic Δ-filtration dimension for such algebras and show that the finitistic dimension for such an algebra is bounded by the sum of the finitistic Δ-filtration dimension and the &∇macr;-filtration dimension. In particular, the finitistic dimension must be finite. We also conjecture that this bound is exact when the algebra has a simple preserving duality. We give several examples of well-known algebras where this is the case, including many of the Schur algebras, and blocks of category O. We also give an explicit combinatorial formula for the global dimension in this case. [ABSTRACT FROM AUTHOR]