Pure Baer injective modules

Pure injective modules over hereditary Noetherian prime rings with enough invertible ideals

In this article, the notion of pure hereditary rings is introduced as a nontrivial generalization of the notion of both pure semisimple rings and of hereditary rings. Some properties and examples of pure hereditary rings are given. For a positive integer n, let 𝒫 n be the class of all left R-modules with projective dimension ≤ n. It is shown that over a left pure hereditary ring is a perfect and hereditary cotorsion pair, which gives a partial positive answer to a question of Göbel and Trlifaj. The modules in the class are also studied. Special attention is paid to the case n = 1.

Abstract.Ziegler’s Indecomposability Criterion is used to prove that a totally transcendental, i.e., ∑-pure injective, indecomposable left module over a left noetherian ring is a directed union of finitely generated indecomposable modules. The same criterion is also used to give a sufficient condition for a pure injective indecomposable module RU to have an indecomposable local dual

We show that over a right coherent ring all pure injective left modules have flat covers. Then using recent work of Auslander and Buchweitz we show that left modules of finite flat dimension over right coherent rings also have flat covers.

There are several notions of size for subsets of a semigroup $$S$$ that originated in topological dynamics and are of interest because of their combinatorial applications as well as their relationship to the algebraic structure of the Stone–Cech compactification $$\beta S$$ of $$S$$ . Among these notions are thick sets, central sets, piecewise syndetic sets, IP sets, and $$\Delta $$ sets. Two related notions, namely $$C$$ sets and $$J$$ sets, arose in the study of combinatorial applications of the algebra of $$\beta S$$ . If the semigroup is noncommutative, then all of these notions have both left and right versions. In any semigroup, a left thick set must be a right $$J$$ set (and of course a right thick set must be a left $$J$$ set). We show here that for any free semigroup or free group on more than one generator, there is a set which satisfies all of the left versions of these notions and none of the right versions except $$J$$ . We also show that for the free semigroup on countably many generators, there is a left $$J$$ set which is not a right $$J$$ set.

Deconstructing Lawvere with distributive laws

This paper consists of three parts. The first is devoted to investigating the equivalence and left-right symmetry of several conditions known to characterize finite dimensional algebras which have a unique minimal faithful representation— QF- 3 3 algebras—in the class of left perfect rings. It is shown that the following conditions are equivalent and imply their right-hand analog: R R contains a faithful Σ \Sigma -injective left ideal, R R contains a faithful I I II -projective injective left ideal; the injective hulls of projective left R R -modules are projective, and the projective covers of injective left R R -modules are injective. Moreover, these rings are shown to be semi-primary and to include all left perfect rings with faithful injective left and right ideals. The second section is concerned with the endomorphism ring of a projective module over a hereditary or semihereditary ring. More specifically we consider the question of when such an endomorphism ring is hereditary or semihereditary. In the third section we establish the equivalence of a number of conditions similar to those considered in the first section for the class of hereditary rings and obtain a structure theorem for this class of hereditary rings. The rings considered are shown to be isomorphic to finite direct sums of complete blocked triangular matrix rings each over a division ring.

Introduction The class of simple first order theories has recently been intensively studied. Simple theories were introduced by Shelah, who attempted to find a strictly broader class than the class of stable theories, in which forking is still reasonably well-behaved. After some years of neglect, the present author proved, in his doctoral thesis, that forking satisfies almost all the basic properties which hold in stable theories. In fact, a large amount of the machinery of stability theory, invented by Shelah, is valid in the broader class of simple theories. All stable theories are simple, but there are also simple unstable theories, such as the theory of the random graph. Many of the theories of particular algebraic structures which have been studied recently (pseudofinite fields, algebraically closed fields with a generic automorphism, smoothly approximable structures turn out to be simple unstable, too. Since a general survey of simple theories written by the author and A. Pillay appeared in the Bulletin of Symbolic Logic, here we shall focus on the notion of canonical base and Zilber's theorem in the context of simple theories. Roughly speaking, Zilber's theorem (Theorem 4.4) on a strongly minimal set says that any nontrivial ω-categorical, strongly minimal set is essentially a vector space over a finite field. Zilber used the notion of canonical base (for stable theories) significantly in the proof of this theorem.

We quickly recall the main concepts of Category Theory with a strong emphasis on the universal property of pullbacks. A whole section is devoted to the art of recognizing them. We define monomorphisms, epimorphisms and split epimorphisms, the last notion being the pillar of our classification process of algebraic structures, via the fibration of points (see the classification table, page 103). Then we investigate the notion of internal equivalence relation and the subtle hierarchy related to the notion of epimorphism. Finally we describe the process of dualization of any categorical notion.

Process theories combine a graphical language for compositional reasoning with an underlying categorical semantics. They have been successfully applied to fields such as quantum computation, natural language processing, linear dynamical systems and network theory. When investigating a new application, the question arises of how to identify a suitable process theoretic model. We present a conceptually motivated parameterized framework for the construction of models for process theories. Our framework generalizes the notion of binary relation along four axes of variation, the truth values, a choice of algebraic structure, the ambient mathematical universe and the choice of proof relevance or provability. The resulting categories are preorder-enriched and provide analogues of relational converse and taking graphs of maps. Our constructions are functorial in the parameter choices, establishing mathematical connections between different application domains. We illustrate our techniques by constructing many existing models from the literature, and new models that open up ground for further development.

We characterize the left Noetherian rings by the existence of decompositions of left modules into direct sums of an injective submodule and a submodule containing no injective submodule except 0. We also prove that a left Noetherian ring is left hereditary iff the suspension of each left ideal (see [7]) is injective or, equivalently, the above decomposition is unique.0. Let A be an Abelian group.Then A = D B, where D is divisible and B is reduced in the sense of Kaplansky [8]; that is, B has no divisible submodule except 0. If this holds for left modules over a unitary ring R, with "injective" and "/-reduced" replacing "divisible" and "reduced", respectively, then R must be left Noetherian.The proof requires the theorem of Eilenberg-Bass (see, e.g., Theorem 20.1, p. Ill, of [2]) which characterizes left Noetherian by the requirement that direct sums of injective left modules are injective.Furthermore, if the decomposition, which will be referred to as an i-decomposition, is unique in the sense that D is the unique injective submodule of A for which A = D ffi B, with B /-reduced, then every sum of injective submodules is injective, and hence R must be left hereditary by a theorem of Matlis [9].At the same time, it will be shown that a left Noetherian ring is left hereditary iff the suspension of each left ideal is injective.

In mathematical physics (such as the one-dimensional time-independent Schrödinger equation), Sturm-Liouville problems occur very frequently. We construct, with a different perspective, a Sturm-Liouville problem in multiplicative calculus by some algebraic structures. Then, some asymptotic estimates for eigenfunctions of the multiplicative Sturm-Liouville problem are obtained by some techniques. Finally, some basic spectral properties of this multiplicative problem are examined in detail.

A logic is presented for reasoning on iterated sequences of formulae over some given base language. The considered sequences, or schemata, are defined inductively, on some algebraic structure (for instance the natural numbers, the lists, the trees etc.). A proof procedure is proposed to relate the satisfiability problem for schemata to that of finite disjunctions of base formulae. It is shown that this procedure is sound, complete and terminating, hence the basic computational properties of the base language can be carried over to schemata.

We present some simple conditions for a module to be injective in certain situations. Injective modules over bialgebras often have the additional structure of module algebras, and our results can be used to give some explicit examples. In particular, we construct such a module algebra for the quantum group O q ( SL ( 2 ) ) {O_q}({\text {SL}}(2)) . Our results can also be used to describe the Hopf algebra duals of O q ( SL ( 2 ) ) {O_q}({\text {SL}}(2)) and related Hopf algebras.

Let S and R be rings, be a semidualizing bimodule and be an integer or We introduce and study C-FP n -injective and C-FP n -flat modules as a common generalization of some known modules such as C-injective (resp. C-FP-injective, C-weak injective) and C-flat (resp. C-projective, C-weak flat) modules. Suppose that is a faithfully semidualizing bimodule. We give some equivalent characterizations of left n-coherent rings in terms of C-FP n -flat left S-modules and C-FP n -injective left R-modules. Then we show that the pairs and are coproduct-closed and product-closed duality pairs and both and are covering and preenveloping, where and denote the classes of C-FP n -flat left S-modules and C-FP n -injective left R-modules respectively. Finally, we investigate Foxby equivalence relative to C-FP n -injective and C-FP n -flat modules.