Adequacy of nonsingular matrices over commutative principal ideal domains

The notion of the adequacy of commutative domains was introduced by Helmer in Bull. Amer.Math. Soc., 49 (1943), 225–236. In the present paper we extend the concept of adequacy to noncommutative Bézout rings. We show that the set of nonsingular second-order matrices over a commutative principal ideal domain is adequate.