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1	Sveikųjų skaičių žiedo plėtinių vieneto dalikliai / Dividers of unit of expansion of a ring of integers Falkevič, Irena 10 June 2004 (has links) 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. Mathematics Plėtinys Unit Ring Expansion Diniders Dalikliai Vienetas Žiedas
2	Stipriai pirminiai moduliai virš žiedų / Strongly prime modules over rings Bandalevičiūtė, Marijana 23 June 2005 (has links) 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. Mathematics Kvaziinjektyvus modulis virš žiedo R Pomodulio esminis plėtinys Stipriai pirminis modulis Annihiliator Homomorphism of the ring R Modulio plėtinys Strongly prime modules over rings Pomodulis Žiedas Žiedų homomorfizmas Ring Žiedo R modulio injektyvus apvalkalas Kairiųjų R-modulių homomorfizmas Idealas Ideal Anuliatorius Faktormodulis Modules

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Sveikųjų skaičių žiedo plėtinių vieneto dalikliai / Dividers of unit of expansion of a ring of integers

Stipriai pirminiai moduliai virš žiedų / Strongly prime modules over rings