Reduction of matrices over orders of imaginary quadratic fields (Q551313)

For square matrices of size 2 by 2, \textit{P. M. Cohn} [Publ. Math., Inst. Hautes Étud. Sci. 30, 365--413 (1966; Zbl 0144.26301)] has used the concept of a standard form which is a very important tool for the investigation of \(GE_2\)-rings. In this paper a special decomposition (called the near standard form) of \((1,2)\)-matrices over a ring is introduced and a method for a reduction of such matrices is explained. This is applied for a detection of elementary second order matrices among invertible second order matrices. The tool is used in detail over orders of imaginary quadratic fields, where an algorithm, a number of properties and examples are presented.