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.

Šiame darbe yra nagrinėjamas piršto antspaudo atpažinimo uždavinio viena iš sudedamųjų dalių: skaitmeninės informacijos apie piršto antspaudą gavimas. Aprašomas metodas, paremtas kreivių kreivumų įvertinimu bei minimalaus žiedo sąvoka. Taip pat, aprašytas naujas Delaunė trianguliacijos radimo, minimalaus žiedo skaičiavimo bei kreivių kreivumų įvertinimo algoritmai. Darbo tikslas - pagerinti piršto antspaudo atpažinimo algoritmo kokybę, bei greitį. / The recognition of fingerprint is discussed in this article. The goal of this work is to increase the quality of fingerprint recognition method, and to improve algorithm speed. The new method of fingerprint data for fingerprint matching is analyzed. It concentrates on calculating values of curve curvatures, and minimum-width annuli. Some new methods of evaluation of this properties are described step by step.

Darbe išanalizuotos pagrindinės žiedų ir modulių lokalizacijos savybės, iš kurių nustatyta, kad modulių lokalizacija yra tiksli, modulis virš žiedo A yra plokščias. Kadangi žiedų lokalizacija dažniausiai nagrinėjama pirminio idealo atžvilgiu, todėl šiame darbe pateiktos pirminių bei maksimalių idealų pagrindinės sąvokos ir teoremos. I��analizuoti tokie teiginiai:  Kiekviename nenuliniame žiede egzistuoja maksimalus idealas.  Bet kuris maksimalus idealas yra pirminis.  Jei p yra žiedo A pirminis idealas, tada Ap yra lokalusis žiedas. Išspręsti tokie uždaviniai: • Tegul Z yra sveikųjų skaičių žiedas ir p yra pirminis skaičius, tada pZ yra pirminis idealas. Be to, pZ yra maksimalus idealas. • Tegul S yra multiplikatyvus žiedo A poaibis ir M yra baigtinai generuotas modulis virš žiedo A, tai modulio loacalizacija lygi 0 tada ir tik tada, kai egzistuoja poaibio S elementas s toks, kad sM=0. / Basic properties of ring and module localization are analyzed. Particularly localization of modules is exact, module over the ring A is flat. Main properties of prime and maximal ideals are stated and analyzed. Particularly, the following propositions are analyzed:  In each ring there exists a maximal ideal.  Each maximal ideal is a prime.  If p is prime ideal of a ring A, then Ap is a local ring. Solved these problems: • If Z is ring of integers and p is a prime number, then pZ is the prime ideal. Also, pZ is a maximal ideal. • If S is multiplicative subset of a ring A and M is a finitely generated module over the ring A, then module localization =0 if and only if when there exists sM=0 with some s.

The purpose of this work is to analyse the analogue of prime modules in commutative case – strongly prime modules over rings in non-commutative case. Strongly prime modules over rings, two-sided and one-sided strongly prime ideals in the rings are examined in the work. Concepts and theorems related to this topic are analysed in the paper. These problems are solved: • Taking the homomorphism of the ring R into ring of endomorphisms of the Abelian group we get all the modules over the ring R. • Annihilators of the nonzero elements of the module over commutative ring coincide and are the prime ideal. • In non-commutative case module is strongly prime only in the case when annihilators its nonzero elements are equivalent. • Finite Cartesian product of strongly prime modules, in which annihilators of the nonzero elements are equivalent, is a strongly prime module.