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Fine rings: A new class of simple rings
A nonzero ring is said to be fine if every nonzero element in it is a sum of a unit and a nilpotent element. We show that fine rings form a proper class of simple rings, and they include properly theExpand
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As a special case of fully invariant subgroups, strongly invariant subgroups are introduced and studied for Abelian groups. 2010 Mathematics Subject Classification. 20K27, 20F99
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In this note, steps in order to write a formula that gives the total number of subgroups of a finite abelian group are made.
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Rings with Lattices of Idempotents
The central idempotents of any ring with identity form a Boolean algebra. This result is largely extended for rings with generalized commuting idempotents.
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A nil-clean 2 × 2 matrix over the integers which is not clean
While any nil-clean ring is clean, the last eight years, it was not known whether nil-clean elements in a ring are clean. We give an example of nil-clean element in the matrix ring ℳ2(Z) which is notExpand
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Lattice concepts of module theory
Preface. List of Symbols. 1. Basic notions and results. 2. Compactly generated lattices. 3. Composition series. Decompositions. 4. Essential elements. Pseudo-complements. 5. Socle. Torsion lattices.Expand
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Hypergroups associated with lattices
Hypergroups associated with modular lattices, respectively compactly generated lattices, are studied and characterized.
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Basic notions and results
Definition. A system (A 1, A 2, ..., A n ; R) with A i 1 ≤ i ≤ n arbitrary sets and R ⊆ A 1 × A 2 × ... × A n is called an n-ary relation between the elements of these sets. If A 1 = A 2 = ... = A nExpand
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Exercises in Abelian Group Theory
Preface. List of Symbols. I: Statements. 1. Basic notions. Direct sums. 2. Divisible groups. 3. Pure subgroups. Basic subgroup. 4. Topological groups. Linear topologies. 5. Algebraically compactExpand
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Compactly generated lattices
Definition (Nachbin, Stenstrom) An element c of a complete lattice L is called compact if for every subset X of L and c ≤ ∨ X there is a finite subset F ⊆ X such that c ≤ ∨ F and S-compact if forExpand