In work expansion of 4 degrees of a ring of integers Z[α] = ({a0 +a1α + a2α2 + a3α3 | a0, a1, a2, a3 ∈ Z}, + , · ) is examined. The problem is solved: what should be integers x0, x1, x2, x3, that ξ = x0 + x1α + x2α2 + x3α3 had opposite in ring Z[α], that is that ξ–1 ∈ Z[α] – that ξ would be a divider of unit. Method of work: we design (basing on linear operators) subring M (m3, m2, m1, m0) of rings of matrixes M[4x4](Z). It subring is isomorphying to ring Z[α]. The fact is used: „ the matrix with the integer elements A ∈ Z[mxm](Z) has an opposite matrix in only case when its determinant |A | = ± 1. From property of isomorphism: “if φ(ξ) = Aξ isomorphism of ring Z[α] in ring M (m3, m2, m1, m0), so (φ(ξ))–1 = φ(ξ–1) = Aξ–1 = (Aξ )–1 “ follows, that ξ ∈ Z[α] has opposite ξ–1 ∈ Z[α] in only case when a matrix Aξ has an opposite matrix. And a matrix Aξ has an opposite matrix in a ring M (m3, m2, m1, m0) in only case when its determinant |A| = ± 1.