Suppose R is a commutative ring with 1, b{cyrillic}=(b{cyrillic}_ij) is a fixed D-net of ideals of R of order n, and Gb{cyrillic} is the corresponding net subgroup of the general linear group GL (n, R). There is constructed for b{cyrillic} a homomorphism det_b{cyrillic} of the subgroup G(b{cyrillic}) into a certain Abelian group Φ(b{cyrillic}). Let I be the index set {1...,n}. For each subset α{subset double equals}I let b{cyrillic}(∝)=∑b{cyrillic}_ijb{cyrillic}_ji, where i, ranges over all indices in α and j independently over the indices in the complement Iα (b{cyrillic}(I) is the zero ideal). Let det_∝(a) denote the principal minor of order |α|≤n of the matrix a ∃ G (b{cyrillic}) corresponding to the indices in α, and let' Φ(b{cyrillic}) be the Cartesian product of the multiplicative groups of the quotient rings R/b{cyrillic}(α) over all subsets α{subset double equals} I. The homomorphism det_b{cyrillic} is defined as follows:[Figure not available: see fulltext.] It is proved that if R is a semilocal commutative Bezout ring, then the kernel Ker det_b{cyrillic} coincides with the subgroup E(b{cyrillic}) generated by all transvections in G(b{cyrillic}). For these R is also defined Tm det_b{cyrillic}.