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Assume that G is a finite group, π(G) = {s} ∪ σ, s > 2, Σ is a set of Sylow σ-subgroups in which one subgroup is taken for each pi 2 σ, and R(G) is the largest normal soluble subgroup in G (soluble radical of G). Moreover, suppose that each Sylow pi-subgroup Gpi 𝜖 Σ normalizes the s-subgroup T(i) ≠ 1 of the group G. In this case, we establish the conditions under which s divides |R(G)|. [ABSTRACT FROM AUTHOR]

We introduce the notion of a ring of almost unit stable rank 1 as a generalization of the notion of a ring of unit stable rank 1. We prove that a ring of almost unit stable rank 1 with nonzero Jacobson radical is a ring of unit stable rank 1 and is a 2-good ring. We introduce the notion of almost 2-good ring and show that an adequate domain is an almost 2-good ring. [ABSTRACT FROM AUTHOR]

We study noncommutative rings in which the Jacobson radical contains a completely prime ideal. It is proved that a right Bézout ring in which the Jacobson radical contains a completely prime ideal is a right Hermite ring. We describe a new class of Bézout rings that are not elementary divisor rings. [ABSTRACT FROM AUTHOR]

It is shown that an adequate ring with nonzero Jacobson radical has a stable range 1. A class of matrices over an adequate ring with stable range 1 is indicated. [ABSTRACT FROM AUTHOR]

We construct Artinian serial rings and tiled orders of width two with maximal finite global dimension. [ABSTRACT FROM AUTHOR]