On the Morita equivalence of idempotent rings and monomorphisms of firm bimodules
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2022-07-06
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Abstract
Selles dissertatsioonis on uuritud idempotentsete ringide Morita ekvivalentsi ning viimases peatükis on täpsemalt vaadeldud erinevat tüüpi bimoodulite kategooriaid. Bimoodulitel on oluline roll Morita teoorias, näiteks esinevad nad Morita kontekstide komponentidena. Ringi nimetatakse idempotentseks, kui iga tema element on esitatav mingite elementide korrutiste summana. Idempotentsed ringid on ühikelemendiga ringide üldistus.
Ilma ühikelemendita ringide Morita ekvivalentsuse defineerimiseks on üldiselt kolm erinevat loomulikku viisi: öelda, et ringid R ja S on Morita ekvialentsed parajasti siis, kui ringide R ja S püsivate, kinniste või unitaarsete-väändeta parempoolsete moodulite kategooriad on ekvivalentsed. Idempotentsete ringide klass on üks suuremaid ringide klasse, kus kõik need viisid omavahel kokku langevad. Lisaks on idempotentsete ringide Morita ekvivalentsi mugav kirjeldada Morita kontekstide abil. Nimelt kehtib tingimus, et idempotentsed ringid R ja S on Morita ekvivalentsed parajasti siis, kui ringide R ja S vahel leidub unitaarne ja sürjektiivne Morita kontekst. See kontekstidega kirjeldus leiab siinses dissertatsioonis rohket kasutust.
Käesoleva dissertatsiooni põhieesmärk on uurida mitmeid algebralisi konstruktsioone, mis on seotud idempotentsete ringide Morita ekvivalentsusega ning nende abil avada idempotentsete ringide Morita ekvivalentsuse mõistet. Lisaks on viimases peatükis erilise vaatluse all just püsivate bimoodulite kategooria ning monomorfismid selles kategoorias.
Antud väitekiri koosneb kuuest peatükist. Esimene peatükk on sissejuhatus, kus antakse lühike ülevaade Morita teooria ajaloost ning seejärel tutvustatakse väitekirja struktuuri.
Teises peatükis on toodud vajalikud eelteadmised, mida läheb vaja, et mõista seda väitekirja. Alustuseks on tutvustatud mõningaid mõisteid kategooriateooriast, nimelt kaasfunktoritega seotud mõisteid ja erinevat liiki monomorfisme. Seejärel on ära toodud vajalikud mõisted ringiteooriast ning moodulite teooriast. Eelteadmiste peatükis on pikemalt tutvustatud ka bimooduleid ning defineeritud erinevad bimoodulite kategooriad. Lõpetuseks on antud Morita teooria algteadmised, s.h. on defineeritud idempotentsete ringide Morita ekvivalentsus ja Morita kontekst ning esitatud Morita ekvivalentsuse kirjeldus kasutades Morita kontekste.
Kolmandas peatükis defineeritakse Reesi-maatriksringi ja tensorkorrutisringi mõisted suvaliste ringide jaoks. Mõlemat konstruktsiooni on edukalt kasutatud, et uurida Morita ekvivalentsust ning on tõestatud tulemus, mis seob omavahel Reesi-maatriksringid ja tensorkorrutisringid. Lisaks on siin peatükis vaadeldud kaas-endomorfismide ringe, millede abil on kirjeldatud s-unitaalsete ringide Morita ekvivalentsus. See peatükk põhineb artiklil [48].
Neljandas peatükis on defineeritud ringide laiendid ning tõestatud mitmeid ringide laiendite lihtsamaid omadusi. Antud peatüki põhiteoreemina on tõestatud, et idempotentsed ringid R ja S on Morita ekvivalentsed parajasti siis, kui leidub nende ringide ühine laiend. Lisaks on seal näidatud, et iga unitaarne ja sürjektiivne Morita kontekst idempotentsete ringide R ja S vahel on isomorfne unitaarse ja sürjektiivse Morita kontekstiga, mis on indutseeritud ringide R ja S ühise laiendi poolt. Lõpetuseks on näidatud, et poolrühmade Morita ekvivalentsus on seotud teatavate ringide ühise laiendiga. Neljas peatükk põhineb artiklil [27].
Viiendas peatükis uuritakse ringi unitaarsete ideaalide kvantaali. Seal on tõestatud, et kui idempotentsed ringid R ja S on Morita ekvivalentsed, siis on R ja S unitaarsete ideaalide kvantaalid isomorfsed. Siin peatükis on seejärel lühidalt uuritud Morita ekvivalentsete ringide sokleid ja nende ringide moodulite annihilaatoreid. Lisaks on tõestatud, et kui kaks ringi on seotud Morita kontekstiga, siis on nende ringide faktorringid vastavate ideaalide järgi samuti seotud sama liiki Morita kontekstiga. Viies peatükk põhineb artiklil [49].
Viimases ehk kuuendas peatükis uuritakse põhjalikult püsivate bimoodulite kategooriat üle mingite idempotentse ringide S ja R. Kõigepealt on siin näidatud, et püsivate, kinniste ja unitaarsete-väändeta (S,R)-bimoodulite kategooriad on tõepoolest ekvivalentsed. Seejärel on kirjeldatud monomorfismid püsivate (S,R)-bimoodulite kategoorias. Lõpetuseks on tõestatud, et mingi püsiva (S,R)-bimooduli M unitaarsete alam-bimoodulite võre on isomorfne bimooduli M (kategoorsete) alamobjektide võrega. Kuues peatükk on artikli [47] üldistus bimoodulite juhule.
The purpose of this thesis is to study the Morita equivalence of idempotent rings using various algebraic constructions. Our goal is to find as many connections as possible between Morita equivalence and the considered constructions. The category of firm bimodules over two idempotent rings and especially monomorphisms in this category will be of special interest. Background The notion of a Morita equivalence for rings with identity first arose in 1958 from the seminal paper [36] by Kiiti Morita. He described when the module categories of two rings with identity are equivalent. Later this situation became known as Morita equivalence of the underlying rings. The resulting Morita theory has proven to be very useful in the development of the theory of rings with identity. First steps for extending Morita equivalence to non-unital rings were made by Abrams in 1983 with [1], who considered rings with local units. Further developments in extending Morita theory to a more wider class of rings were made by Komatsu in 1986 with [20] for s-unital rings and Ành and Màrki in 1987 with [7] for rings with local units. Later in 1991 Garcia and Simòn developed Morita theory for idempotent rings in [14]. One especially useful tool -- which is widely used in this thesis -- for studying Morita equivalence is the notion of a Morita context. Morita contexts were first introduced by Bass in 1962 in [8], who called them preequivalence datas. They were extensively used by Amitsur in 1971 in [3] and Müller in 1972 in [37] and have become increasingly popular for studying Morita equivalence ever since. Morita equivalence is defined using the equivalence of certain module categories. This makes it obvious that it is an equivalence relation on the class of rings, but on the other hand equivalence functors are hard to work with, especially if we wish to understand the structure of Morita equivalent rings. Morita contexts are helpful here, because they are much more concrete objects. Essentially, they consist of two bimodules and two bimodule homomorphisms. Over the years, Morita theory has also been developed for many different algebraic structures, e.g. monoids (by Banaschewski and Knauer), semigroups (by Talwar in 1995 with [43]), quantales, C*-algebras etc. Morita equivalence of semigroups will be of particular interest in this thesis, because we will introduce several notions used to study Morita equivalence of semigroups into the ring case. In particular enlargements borrowed from Lawson's article [29] and strict local isomorphisms borrowed from Màrki's and Steinfeld's paper [35]. Although the Morita theory of factorizable semigroups and idempotent rings are similar in some aspects, there exist some considerable differences. For instance, if two monoids are Morita equivalent, then either of them is an enlargement of the other, but two Morita equivalent rings with identity need not be isomorphic to their joint enlargement. Also we will show that the only idempotent ring Morita equivalent to {0} is {0} itself. This is a considerable difference from the Morita equivalence of semigroups, because there are many infinitely semigroups Morita equivalent to the one-element semigroup. Finally, we will thoroughly study the category of firm bimodules over idempotent rings. The term ``firm module'' was first used by Quillen in 1996 in [39]. Although, a similar notion for modules over unital algebras was already introduced by Taylor in 1982 in [45] under the name regular modules. Categories of firm modules and their applications in Morita theory have been extensively studied by Marin in his master's thesis [33] and [34] in 1998 and later with Garcia and Gonzàlez-Fèrez in articles [12], [13], [17] and [18]. Overview of the thesis This thesis is divided into six chapters. The first chapter is the introduction, where we give a short historical overview of developments in Morita theory. Subsequently the summary of the thesis is presented. In Chapter 2 we will give the preliminaries, which are necessary for understanding the material of this thesis. We will try to keep the text rather self-contained. First, we will introduce some notions from category theory, which will be used in what follows. Namely we define adjoint functors and several kinds of monomorphisms. Next, we present the basics of ring and module theory and after that introduce bimodules. Finally, we will introduce Morita theory by defining and describing Morita equivalence for idempotent rings and Morita contexts. In Chapter 3 we will define Rees matrix rings and tensor product rings for arbitrary rings. We will use both of these concepts to study Morita equivalence of idempotent rings. It turns out that every idempotent Rees matrix ring is Morita equivalent to its ground ring (Theorem 3.8). We define pseudo-surjective mappings. We see that every tensor product ring over an idempotent ring R, which is defined by a pseudo-surjective (R,R)-bilinear map, is Morita equivalent to R (Theorem 3.16). Then we define strict local isomorphisms of rings, inspired by a similar notion in semigroup theory introduced by Màrki and Steinfeld. We show that if two rings are Morita equivalent, then any pseudo-surjectively defined tensor product ring over one of those rings is strictly locally isomorphic to the other one (Corollary 3.24). Finally, we prove a result connecting the constructions of Rees matrix rings and tensor product rings (Theorem 3.40). We will also study the rings of adjoint endomorphisms of modules. This approach is a generalization of the ideas used by Ành in [5]. We use adjoint endomorphisms to describe Morita equivalence of s-unital rings (Theorem 3.39). This section is based on [48]. In Chapter 4 we will define enlargements of rings, which is again a notion borrowed from semigroup theory. First, we prove some simple properties of enlargements and then give two natural constructions that produce enlargements. We will show that enlargements -- namely the existence of a joint enlargements -- can be used to describe Morita equivalence of idempotent rings (Theorem 4.13). For instance, this description allows us to easily conclude that the only ring Morita equivalent to {0} is {0} itself (Corollary 4.15). Furthermore, we will show that for any two Morita equivalent idempotent rings there exists a Morita context between those rings, where the bimodules are induced by their joint enlargement (Corollary 4.21). Finally, we show that a joint enlargement of certain particular rings is lurking behind the strong Morita equivalence of semigroups (Theorem 4.25). This section is based on [27]. In Chapter 5 we will study unitary ideals of Morita equivalent idempotent rings. First, we show that the set of all unitary ideals of an idempotent ring actually forms a unital quantale (Proposition 5.3). In particular we will prove that that the quantales of unitary ideals of Morita equivalent idempotent rings are isomorphic (Theorem 5.8). Next, we will briefly consider socles and annihilators in connection to Morita equivalence. Finally, we will prove that if two idempotent rings are Morita equivalent, then their quotients, by the ideals that correspond to each other, are also Morita equivalent (Theorem 5.16). Essentially, we will give a way of factorizing Morita contexts by ideals. This section is based on [49]. In Chapter 6 we will study the category of firm bimodules over two idempotent rings. First, we will have a lengthy detour concerning the subcategories of firm, closed and torsion-free bimodules over idempotent rings. Due to the size of this section it is divided into subsections. After introducing the categories of firm and closed bimodules over some idempotent rings, we will show explicitly that these categories are equivalent to each other and also to the category of unitary torsion-free bimodules over the same rings (Theorem 6.14). Moreover, the category of closed bimodules over some idempotent rings is an essential localization of the category of all bimodules over those rings (Theorem 6.16). Next, we will describe monomorphisms in the category of all bimodules and in the category of all unitary bimodules over two rings. There we will also give an example of a non-injective monomorphism in the category of unitary bimodules over some particular rings, proving that this category is not balanced (Example 6.23). Finally, we will describe monomorphisms in the category of firm bimodules (Theorem 6.25) and show that the lattice of unitary sub-bimodules of a given firm bimodule is isomorphic to the lattice of categorical subobjects of this bimodule (Theorem 6.29). This chapter is a generalization of [47].
The purpose of this thesis is to study the Morita equivalence of idempotent rings using various algebraic constructions. Our goal is to find as many connections as possible between Morita equivalence and the considered constructions. The category of firm bimodules over two idempotent rings and especially monomorphisms in this category will be of special interest. Background The notion of a Morita equivalence for rings with identity first arose in 1958 from the seminal paper [36] by Kiiti Morita. He described when the module categories of two rings with identity are equivalent. Later this situation became known as Morita equivalence of the underlying rings. The resulting Morita theory has proven to be very useful in the development of the theory of rings with identity. First steps for extending Morita equivalence to non-unital rings were made by Abrams in 1983 with [1], who considered rings with local units. Further developments in extending Morita theory to a more wider class of rings were made by Komatsu in 1986 with [20] for s-unital rings and Ành and Màrki in 1987 with [7] for rings with local units. Later in 1991 Garcia and Simòn developed Morita theory for idempotent rings in [14]. One especially useful tool -- which is widely used in this thesis -- for studying Morita equivalence is the notion of a Morita context. Morita contexts were first introduced by Bass in 1962 in [8], who called them preequivalence datas. They were extensively used by Amitsur in 1971 in [3] and Müller in 1972 in [37] and have become increasingly popular for studying Morita equivalence ever since. Morita equivalence is defined using the equivalence of certain module categories. This makes it obvious that it is an equivalence relation on the class of rings, but on the other hand equivalence functors are hard to work with, especially if we wish to understand the structure of Morita equivalent rings. Morita contexts are helpful here, because they are much more concrete objects. Essentially, they consist of two bimodules and two bimodule homomorphisms. Over the years, Morita theory has also been developed for many different algebraic structures, e.g. monoids (by Banaschewski and Knauer), semigroups (by Talwar in 1995 with [43]), quantales, C*-algebras etc. Morita equivalence of semigroups will be of particular interest in this thesis, because we will introduce several notions used to study Morita equivalence of semigroups into the ring case. In particular enlargements borrowed from Lawson's article [29] and strict local isomorphisms borrowed from Màrki's and Steinfeld's paper [35]. Although the Morita theory of factorizable semigroups and idempotent rings are similar in some aspects, there exist some considerable differences. For instance, if two monoids are Morita equivalent, then either of them is an enlargement of the other, but two Morita equivalent rings with identity need not be isomorphic to their joint enlargement. Also we will show that the only idempotent ring Morita equivalent to {0} is {0} itself. This is a considerable difference from the Morita equivalence of semigroups, because there are many infinitely semigroups Morita equivalent to the one-element semigroup. Finally, we will thoroughly study the category of firm bimodules over idempotent rings. The term ``firm module'' was first used by Quillen in 1996 in [39]. Although, a similar notion for modules over unital algebras was already introduced by Taylor in 1982 in [45] under the name regular modules. Categories of firm modules and their applications in Morita theory have been extensively studied by Marin in his master's thesis [33] and [34] in 1998 and later with Garcia and Gonzàlez-Fèrez in articles [12], [13], [17] and [18]. Overview of the thesis This thesis is divided into six chapters. The first chapter is the introduction, where we give a short historical overview of developments in Morita theory. Subsequently the summary of the thesis is presented. In Chapter 2 we will give the preliminaries, which are necessary for understanding the material of this thesis. We will try to keep the text rather self-contained. First, we will introduce some notions from category theory, which will be used in what follows. Namely we define adjoint functors and several kinds of monomorphisms. Next, we present the basics of ring and module theory and after that introduce bimodules. Finally, we will introduce Morita theory by defining and describing Morita equivalence for idempotent rings and Morita contexts. In Chapter 3 we will define Rees matrix rings and tensor product rings for arbitrary rings. We will use both of these concepts to study Morita equivalence of idempotent rings. It turns out that every idempotent Rees matrix ring is Morita equivalent to its ground ring (Theorem 3.8). We define pseudo-surjective mappings. We see that every tensor product ring over an idempotent ring R, which is defined by a pseudo-surjective (R,R)-bilinear map, is Morita equivalent to R (Theorem 3.16). Then we define strict local isomorphisms of rings, inspired by a similar notion in semigroup theory introduced by Màrki and Steinfeld. We show that if two rings are Morita equivalent, then any pseudo-surjectively defined tensor product ring over one of those rings is strictly locally isomorphic to the other one (Corollary 3.24). Finally, we prove a result connecting the constructions of Rees matrix rings and tensor product rings (Theorem 3.40). We will also study the rings of adjoint endomorphisms of modules. This approach is a generalization of the ideas used by Ành in [5]. We use adjoint endomorphisms to describe Morita equivalence of s-unital rings (Theorem 3.39). This section is based on [48]. In Chapter 4 we will define enlargements of rings, which is again a notion borrowed from semigroup theory. First, we prove some simple properties of enlargements and then give two natural constructions that produce enlargements. We will show that enlargements -- namely the existence of a joint enlargements -- can be used to describe Morita equivalence of idempotent rings (Theorem 4.13). For instance, this description allows us to easily conclude that the only ring Morita equivalent to {0} is {0} itself (Corollary 4.15). Furthermore, we will show that for any two Morita equivalent idempotent rings there exists a Morita context between those rings, where the bimodules are induced by their joint enlargement (Corollary 4.21). Finally, we show that a joint enlargement of certain particular rings is lurking behind the strong Morita equivalence of semigroups (Theorem 4.25). This section is based on [27]. In Chapter 5 we will study unitary ideals of Morita equivalent idempotent rings. First, we show that the set of all unitary ideals of an idempotent ring actually forms a unital quantale (Proposition 5.3). In particular we will prove that that the quantales of unitary ideals of Morita equivalent idempotent rings are isomorphic (Theorem 5.8). Next, we will briefly consider socles and annihilators in connection to Morita equivalence. Finally, we will prove that if two idempotent rings are Morita equivalent, then their quotients, by the ideals that correspond to each other, are also Morita equivalent (Theorem 5.16). Essentially, we will give a way of factorizing Morita contexts by ideals. This section is based on [49]. In Chapter 6 we will study the category of firm bimodules over two idempotent rings. First, we will have a lengthy detour concerning the subcategories of firm, closed and torsion-free bimodules over idempotent rings. Due to the size of this section it is divided into subsections. After introducing the categories of firm and closed bimodules over some idempotent rings, we will show explicitly that these categories are equivalent to each other and also to the category of unitary torsion-free bimodules over the same rings (Theorem 6.14). Moreover, the category of closed bimodules over some idempotent rings is an essential localization of the category of all bimodules over those rings (Theorem 6.16). Next, we will describe monomorphisms in the category of all bimodules and in the category of all unitary bimodules over two rings. There we will also give an example of a non-injective monomorphism in the category of unitary bimodules over some particular rings, proving that this category is not balanced (Example 6.23). Finally, we will describe monomorphisms in the category of firm bimodules (Theorem 6.25) and show that the lattice of unitary sub-bimodules of a given firm bimodule is isomorphic to the lattice of categorical subobjects of this bimodule (Theorem 6.29). This chapter is a generalization of [47].
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algebra, algebraic structures, rings (math.), ring theory, modules (math.), semigroups (math.)