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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]