Coimbra Mathematical Texts
Volume 1 / 2021
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Chapter and Conference Paper
Pre-Lie Algebras, Their Multiplicative Lattice, and Idempotent Endomorphisms
We introduce the notions of pre-morphism and pre-derivation for arbitrary non-associative algebras over a commutative ring k with identity. These notions are applied to the study of pre-Lie k-algebras and, more g...
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Article
Open AccessSpectra of Groups
The aim of this paper is to investigate the behaviour of prime and semiprime subgroups of groups, and their relation with the existence of abelian normal subgroups. In particular, we study the set Spec(G) of all ...
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Article
Open AccessFree Covers and Minimal Sets of Generators
We continue our study of the relation between existence of projective covers and existence of minimal sets of generators for right modules over an arbitrary ring.
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Article
Abstractly constructed prime spectra
The main purpose of this paper is a wide generalization of one of the results abstract algebraic geometry begins with, namely of the fact that the prime spectrum
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Article
Open AccessProjective covers over local rings
We describe the structure of the projective cover of a module \(M_R\) M ...
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Book Series
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Book
New Perspectives in Algebra, Topology and Categories
Summer School, Louvain-la-Neuve, Belgium, September 12-15, 2018 and September 11-14, 2019
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Chapter
Commutative Monoids, Noncommutative Rings and Modules
These are the notes of a non-standard course of Algebra. It deals with elementary theory of commutative monoids and non-commutative rings. Most of what is taught in a master course of Commutative Algebra holds...
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Article
Homomorphisms with Semilocal Endomorphism Rings Between Modules
We study the category Morph(Mod-R) whose objects are all morphisms between two right R-modules. The behavior of the objects of Mod-R whose endomorphism ring in Morph(Mod-R) is semilocal is very similar to the beh...
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Article
Covering classes, strongly flat modules, and completions
We study some closely interrelated notions of Homological Algebra: (1) We define a topology on modules over a not-necessarily commutative ring R that coincides with the R-topology defined by Matlis when R is comm...
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Article
Pretorsion theories, stable category and preordered sets
We show that in the category of preordered sets there is a natural notion of pretorsion theory, in which the partially ordered sets are the torsion-free objects and the sets endowed with an equivalence relatio...
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Book and Conference Proceedings
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Chapter and Conference Paper
What is the Spectral Category?
For a category \(\mathcal {C}\) with finite limits and a class \(\mathcal {S}\) of monomorphisms in \(\mathcal {C}\) that is pullback stable, contains all isomorphisms, is closed under composition, and has the st...
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Article
Factorizations of polynomials with integral non-negative coefficients
We study the structure of the commutative multiplicative monoid \(\mathbb {N}_0[x]^*\) ...
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Book
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Chapter
Some Open Problems
In this final chapter, we pose some questions and present some open problems. Some of them are classical problems.
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Chapter
Monoids, Krull Monoids, Large Monoids
In this chapter, we review what we will need in the rest of the book as far as commutative monoids are concerned. This will show how much we assume of the reader. The contents of Sections 1.5 and 1.7 are excep...
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Chapter
Semilocal Rings
In this section, we recall the notions of Goldie dimension of a modular lattice and Goldie dimension of a module. In fact, the notion of Goldie dimension of a module concerns the lattice L(M) of all submodules...
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Chapter
Spectral Category and Dual Construction
Now we will present a construction due to [Gabriel and Oberst]. Recall that a Grothendieck category is an abelian category with arbitrary coproducts, with exact direct limits and with a generator. Every Grothendi...
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Chapter
The Krull–Schmidt Theorem in the Case Two
In the first part of this chapter, we will describe what occurs if in the Krull– Schmidt–Azumaya theorem (Theorem 4.34), instead of considering modules with local endomorphism rings, that is, endomorphism ring...
