On Hulls of Semiprime Rings

By

Caroline Namanya BSc(ED)(Mak)

2016/HD13/498U

A Dissertation Submitted to the Directorate of Research and Graduate Training in Partial Fulllment of the Award of the Degree of Master of Science in Mathematics of Makerere University

May, 2018

Declaration

I Caroline Namanya, here by declare that this my original work and it has never been

submitted for any Examination.

Signed: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Date: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Approval

This dissertation entitled: On Hulls of Semiprime Rings , has been submitted for exam-

ination with the approval of the following supervisors:

1. Dr. D. Ssevviiri,

Department of Mathematics,

Makerere University.

Signed: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Date: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. Dr. A. S. Bamunoba

Department of Mathematics,

Makerere University.

Signed: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Date: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii

Dedication

This dissertation is dedicated to my parents Mr. Fostino. T . Ahimbisibwe and Mrs. Har-

riet.T. Kyohirwe. May the Almighty reward you abundantly.

iii

Acknowledgement

I am entirely thankful to the Almighty who has given me a healthy life and enabled to

complete this pro ject. I am very grateful to my supervisors; Dr. D. Ssevviiri and Dr. A.

S. Bamunoba, thank you for your unconditional guidance and help. To all lecturers in the

Mathematics department of Makerere university, I am very thankful for teaching me.

I am very thankful to my parents who never gave up on me and my brothers and sisters who

inspire me endlessly. My special friend Ivan Sseguya, no amount of words could ever express

my gratitude towards you. I thank my classmates Hellen Nanteza, George William Luyinda,

Brian Makonzi,Moses sengendo, Walter Okongo, Francis Mugabi, Cosmas Muhumuza and

Samuel Kinen for being my academic friends. I thank my friends Esther Nabimanya, Mary

Jacinta Adjiro, Haroonah Kibirige and Jafali Lutaaya for always being my friends no mater

what.

iv

Contents

Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i

Approval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iii

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iv

Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .v

List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .viii

1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

1.2 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

1.3 Ob jectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 1.3.1 Main ob jective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

1.3.2 Specic ob jectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

1.3.3 Methodolgy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

1.4 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4

1.4.1 Injective hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4

1.4.2 The maximal right ring of quotients of R. . . . . . . . . . . . . . . .5

1.4.3 FI-extending property . . . . . . . . . . . . . . . . . . . . . . . . . .5

1.4.4 Baer and quasi-Baer properties . . . . . . . . . . . . . . . . . . . . .7

1.4.5 Self-injectivity of rings . . . . . . . . . . . . . . . . . . . . . . . . . .8

v

2 Preliminaries 9

2.1 General ring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 2.1.1 Baer, Quasi-Baer and FI-extending rings . . . . . . . . . . . . . . . .12

2.1.2 Prime ideals, prime rings and semiprime rings . . . . . . . . . . . . .14

2.1.3 Radicals and krull dimensions . . . . . . . . . . . . . . . . . . . . . .15

2.2 Rings of quotients of R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16

2.3 General module theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

2.3.1 Modules and submodules . . . . . . . . . . . . . . . . . . . . . . . . .21

2.3.2 Injective modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

2.3.3 Rational and essential extensions . . . . . . . . . . . . . . . . . . . .27

2.4 Ring hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32

3 Constructions of the maximal right ring of quotients of R38

3.1 Q(R ) in terms of the rational hull . . . . . . . . . . . . . . . . . . . . . . . .38

3.2 Q(R) in terms of dense right ideals . . . . . . . . . . . . . . . . . . . . . . .41

4 Constructions of the intermediate ring between Rand its essential over-

rings. 47

4.1 The intermediate ring RB (Q (R )) . . . . . . . . . . . . . . . . . . . . . . . . .47

5 Transference of properties between a semiprime ring Rand its overrings 55

5.1 Transference of properties between Rand RB (Q (R )) . . . . . . . . . . . . . .55

References 61

vi

List of symbols

vii

Abstract

viii

Chapter 1

Introduction

1.1 Background

Extending notions about rings (resp. modules) to their overrings (resp. overmodules) is a

crucial sub ject of study in the theory of rings and modules. In algebra, to study an ob ject,

it is important to search for an overob ject. This overob ject should: belong to a class with

desirable properties, be clearly constructed and allow transference of properties between the

base ob ject and the overob ject.

In this pro ject, we discuss transference of properties between a semiprime unital ring Rand

its essential extensions. A semiprime ring is discussed as a generalization of prime rings and

we also discus prime ideals. The intersection of prime ideals of a ring is its prime radical. A

semiprime ring can be described as a ring with a zero prime radical.

In Chapter 2 of this pro ject, we give denitions of Baer, quasi-Baer and Fully Invariant (FI)-

extending rings. The notion of quasi-Baer rings was dened by (Clark, 1967). (Pollingher &

Zaks, 1970) proves that unlike the class of Baer rings, the class of quasi-Baer rings is closed

under full and triangular matrix extensions. (Birkenmeier, Park, & Rizvi, 2006) notes that

when Ris semiprime, the quasi-Baer and FI-extending ring properties coincide. Overrings

of the ring that are Baer, quasi-Baer and FI-extending rings are discussed in Chapter 4.

In Section 2.3 of Chapter 2 we discuss general module theory. In the same chapter, we dene

injective modules. Injective modules are unavoidable if we are to talk about the injective

hull of a module. The notion of injective modules was rst dened by (Baer, 1940) where he

describes a module to be injective if and only if it is a direct summand of its submodules.

In this pro ject we dene injective modules using Baer’s criterion and in relation to functors

of categories of modules.

1

We then discuss rational and essential extensions of modules. Here we discuss the rational

hull (using dense submodules of a module) and the injective hull (rst described in (Eckmann

; Schopf, 1953)) using essential submodules of a module. Essential extensions of a module

give a new description of injectivity of module. The relationship between essential and

rational extensions of a module are also discussed.

In Chapter 3 of the pro ject, we use the rational hull of a module to describe the maximal ring

Q (R ) of the right rings of quotients of R( right rings of quotients are discussed in Chapter

1). We also discuss another construction of the maximal ring of right ring of quotients

using dense ideals of a ring R. We realize that this maximal right ring of quotients is an

R -submodule of an injective hull of a right R-module.

There is sucient literature that shows that there are dierences in the properties of a ring

and those of its essential extensions such as the injective hull and Q(R ). we construct an

intermediate ring between Rand Q(R ). The intermediate ring will allow transference of

properties between the base ring and its self. This intermediate ring is a subring of Q(R )

generated by Rand the central idempotents of Q(R ) (a subset of the injective hull).

We then discuss the properties transferred between a semiprime ring Rand this essential

overring of R. we also discuss properties such as dierent types of regularities a ring and index

of nilpotency of a ring. We present a counter example showing that in general information

does not always transfer between Rand Q(R ). This shows the need of an intermediate ring

between the base ring Rand its essential overrings to be able to transfer information between

the two.

1.2 Statement of the problem

Generally, a ring Rand its essential extensions such as the maximal right ring of quotients

Q (R ) and the injective hull E(R

R)

have disparities between them. These disparities make it

hard for the transference of properties between the ring Rand these essential extensions.

1.3 Ob jectives

1.3.1 Main ob jective

To obtain the intermediate ring between a semiprime ring Rand its essential overring Q(R )

that allows transference of properties between Rand Q(R ).

2

1.3.2 Specic ob jectives

(i)To give dierent constructions of the maximal right ring of quotients Q(R ) and its

properties.

(ii)To construct the intermediate ring RB (Q (R )) between a semiprime ring Rand Q(R )

that is an optimal choice among the right essential overrings of Rwhich allows trans-

ference of properties.

(iii)To prove the transference of specic properties between a semiprime ring Rand RB (Q (R )) .

(iv)To construct counter examples to demonstrate that in general, the transference of properties between Rand Q(R ) fails.

1.3.3 Methodolgy

The rst specic ob jective is achieved by proving Theorem 1.3.9, (Birkenmeier et al.,

2013) and Theorem 13.21, (Lam, 1999). The proof of the former shows that Q(R ) can

be constructed taking the endomorphism ring on the injective hull and then identifying

the rational hull with Q(R ). The proof of the latter shows that Q(R ) can be dened

as a ring whose elements are classes of R-homomorphisms of dense ideals of Rinto R.

To achieve the second ob jective, we put in use the theory of ring hulls dened in

Denitions 2.1 and 1.6 of (Birkenmeier et al., 2006) which describe the right ring hulls

and the pseudo right ring hulls of a ring. By using the fact that B(Q (R )) Ta right

FI-extending right ring of quotients of Rwe show that the smallest quasi-Baer ring of

quotients RB (Q (R )) = the smallest FI-extending ring of quotients of R, ^

Q FI (

R ). By

use of a proposition which states that, let Rbe a semiprime ring, then Ris FI-extending

if and only if Ris quasi-Baer if and only if Ris essentially quasi-Baer if and only if for

any ideal Iof Rthere is e2 B(R ) such that the R-module I

R is essential in an

R-module

eR Rgenerated by

e. It is shown that RB (Q (R )) = ^

Q qB (

R ) which yields to RB (Q (R ))

= the FI-extending pseudo hull with respect to Q(R ), R(FI ,Q (R )) . Furthermore, we

show that if Ris reduced then Q

qB (

R ) = Q

B(

R ) which is the Baer absolute right ring

that is reduced. Hence using the fact that reduced rings are semiprime, then ^

Q qB (

R )

= Q

qB (

R ) which is the minimal element of the class of quasi-Baer rings (right ring

hull).

To achieve the third specic ob jective, we use a corollary which states that, for a ring

R , the following are equivalent;

(i) Ris von Neumann regular.

3

(ii)

RB (Q (R )) is von Neumann regular.

(iii) Ris semiprime and ^

Q qB (

R ) is von Neumann regular.

We also show that Ris von Neumann regular if and only if RB (Q (R )) is von Neu-

mann regular, hence achieving our specic ob jective by using the fact that RB (Q (R ))

= ^

Q qB (

R ) = RB (Q (R )) when Ris semiprime. Additionally, using Proposition 4 in J.

Hannah’s work on quotient rings of semiprime rings with bounded index which con-

cludes that a semiprime ring Rwith a bounded index is left and right singular. Hence,

^

Q qB (

R ) = Q

qB (

R ). We then show that a semiprime ring Rhas a bounded index of

nilpotency of at most nif and only if Q

qB (

R ) = RB (Q (R )) has a bounded index of

nilpotency of at most n.

To achieve the fourth specic ob jective, we consider an example of a group ring ZG of

the group G=f1, gg over the ring Z. With a conclusion that ZG is not von Neumann

regular while Q(G ) the right ring of quotients of Gis von Neumann regular, hence

showing that information does not transfer between Q(G )and ZG .

1.4 Literature review

1.4.1 Injective hull

In their work on injective hulls of group rings, K. A. Brown and J. Lawrence, (1979), use

the fact that the maximal right ring of quotients Q(R ) of a ring Ris a right R-submodule of

the right injective hull E(R

R)

of Rto ascertain when the maximal right ring of quotient of

the group algebra kGis a right self-injective ring. According to their work the conditions

for this to happen are that kGis right nonsingular and Q(kG )is also injective when Gis

nite.

Basing on Osofsky’s results that showed that the right injective hull E(R

R)

of a ring R, in

general, does not have a ring multiplication which extends its R-module multiplication by

using the ring R=

Z4 2

Z

4

0 Z

4

, where Z

4 is the ring of integers modulo 4. G. F. Birkenmeier

et al., (2009) explicitly characterize all right essential overrings of this ring R.They investigate

various properties as well as interrelationships between these right essential overrings. They

further determine various ring hulls for the ring Rsuch as the right FI-extending, right

extending, and right self-injective ring hulls. Moreover, they nd an intermediate R-module

S R between

R

R and

E(R

R)

which has one compatible ring structure that is right self-injective.

This very much emphasizes the need to use a class of rings that generalizes self injective rings

4

in transference of properties between a ring and its essential overrings.

Although the every module has an injective hull, it is generally hard to construct or describe

it. However certain known subsets of the injective hull can be used to generate an overring

in conjunction with the base ring to serve as a hull of the ring with some desirable proper-

ties. These overrings are close enough to the base ring to facilitate an eective transfer of

information between the base ring and the overrings (Birkenmeier et al., 2013).

In general, the injective hull may not admit any ring structure extending the R-module

structure on E(R

R)

. However, the fact that an R-submodule of E(R

R)

, that is to say Q(R )

has a natural ring structure is a valuable piece of information when transferring properties

between a ring and its essential overrings (Lam, 1999).

1.4.2 The maximal right ring of quotients of R

In the investigations of G. F. Birkenmeier et al., (2008) on connections between the right

FI-extending right ring hulls of semiprime homomorphic images of a ring Rand the right FI-

extending right rings of quotients of Rby considering ideals of Rwhich are essentially closed

and contain the prime radical P(R ). It is stated that semiprime rings exhibit optimal behavior

with respect to the existence (and uniqueness) of right FI-extending right ring hulls and the

transference of various interesting properties between the ring and its right FI-extending ring

hull. Moreover, for an arbitrary semiprime ring R, it seems natural to investigate connections

between the right FI-extending right ring hulls Rand the right FI-extending right rings of

quotients of R.

In another construction of Q(R ), the proofs of Proposition 13.8, (Lam, 1999) and Theorem

1.3.9, (Birkenmeier et al., 2013) clearly show that the maximal right ring of quotients Q(R )

of Rcan be identied with the rational hull of R

R. This equivalence gives the rational hull

a ring structure extending its given right R-module structure. Moreover, it is shown in

Theorem 13.11, (Lam, 1999) that Q(R ) is a ring of quotients of Rwhich is maximal.

1.4.3 FI-extending property

By restricting to the class of semiprime rings, its possible to show that if Ris a semiprime

ring then RB(Q (R )) is both a quasi-Baer right ring hull and a right FI-extending ring hull,

since the right FI-extending property extends to all right essential overrings of a ring. This

allows us to conclude that any right essential overring of a semiprime ring Rthat contains

B (Q (R )) must be quasi-Baer and right FI-extending. Additionally, for a semiprime ring R,

5

the symmetric ring of quotients, the Martindale right ring of quotients, and

Q(R ) are all

quasi-Baer and right FI-extending overrings of R(Birkenmeier, Park, ; Rizvi, 2009b).

It is shown that every nitely generated pro jective module P

R over a semiprime ring

Rhas

the smallest FI-extending essential module extension called the absolute FI-extending hull

of P

R in a xed injective hull of

P

R. Moreover, it is shown that a nitely generated pro jective

module P

R over a semiprime ring

Ris FI-extending if and only if it is a quasi-Baer module

and if and only if End( P

R) is a quasi-Baer ring (Birkenmeier et al., 2009c). Thus this

emphasizes the relatedness of FI-extending and quasi-Baer essential extensions when the

ring Ris semiprime.

In his work on FI-extending modules, K. Ming (2003) brings to light the strength of the

FI-extending property over the extending property of rings. He clearly observes that direct

sum of extending modules and triangular matrix rings over right extending rings are not

necessarily extending. However the FI-extending property is preserved under these various

ring extensions.

The paper titled generalized triangular matrix rings and the fully invariant extending prop-

erty by G. F. Birkenmeier et al., (2002), in which they fully characterize the 2 by 2 generalized

(or formal) triangular matrix rings which are either (right) FI-extending, (right) strongly FI-

extending or quasi-Baer. It is stated that for semiprime rings the FI-extending condition,

strongly FI-extending condition and quasi-Baer conditions are equivalent.

It is made clear in G. F. Birkenmeier’s et al., (2009) work that in general, the right ring hulls

and pseudo right ring hulls are distinct and not unique if they exist even if the ring is right

nonsingular. With examples it is shown that there is a quasi-Baer ring with non isomorphic

right FI-extending right ring hulls. It is also shown that there is a ring with all its right

FI-extending right ring hulls that are mutually isomorphic but does not have a quasi-Baer

right ring hull. However, the condition that a ring is semiprime is a solution to this chaotic

situation.

In their paper titled modules in which every fully invariant submodule is essential in a di-

rect summand, G. F. Birkenmeier et al., (2002), show that unlike the extending property,

the FI-extending property carries over to matrix rings. They go ahead to investigate the

interconnections between the FI-extending property and other related conditions such as

extending and quasi-Baer properties. They further point out that for these rings, the con-

ditions become tighter in the presence of the semiprime, right nonsingular and compliment

bounded conditions on the rings.

6

1.4.4 Baer and quasi-Baer properties

The Baer ring property and the Rickart ring property do not transfer from a ring Rto two

of its important ring extensions, namely, the matrix rings and the polynomial rings over

R . The diculties in these cases motivate the need to study classes of rings for which such

transfers can take place easily even under somewhat weaker conditions. This brings us to

the notion of quasi-Baer rings where one studies a generalized Baer property in which the

annihilators of ideals instead of nonempty subsets of the rings are generated by idempotents

as ideals (Birkenmeier et al., 2013).

One of the results of G. F. Birkenmeier et al., (2000) on quasi-Baer ring extensions and

biregular rings is that Ris a semiprime quasi-Baer ring if Sis a semiprime quasi-Baer

subring of Rsuch that the set of central idempotents of Sis contained in the set of central

idempotents of Rand every nonzero ideal of Rhas nonzero intersection with S. The work also

answers these two questions namely; If the centre of a ring Ris Baer when is Rquasi-Baer?

And More generally, when does the quasi-Baer condition extend to a ring from a subring?

It is rst proved that for a ring Rand a group G, if a group ring RGis quasi-Baer then so is

R and if in addition Gis nite then jG j-

1

2 R, where jG j-

1

is the inverse of the order of the

group G. Counter examples are then given to answer Hirano’s question which asks whether

the group ring RGis quasi-Baer if Ris quasi-Baer and Gis a nite group with jG j-

1

2 R.

Furthermore, eorts are made towards answering the question of when the group ring RG

of a nite group Gis quasi-Baer, and various quasi-Baer group rings are identied (Yi ;

Zhou, 2007). This shows the well behavedness of the quasi-Baer property a ring in relation

to extensions of rings, in particular, to group rings.

In Example 4.16, (Birkenmeier et al., 2010), it is shown that there is a quasi-Baer ring

R (hence Ritself is a quasi-Baer right ring hull of R), but Rdoes not have a unique right

FI-extending right ring hull. In fact, there is a ring Rwhich has mutually isomorphic right

FI-extending ring hulls, but Rhas no quasi-Baer right essential overring Example 4.17,

(Birkenmeier et al., 2010). By Theorem 4.18, (Birkenmeier et al., 2010), the existence

and uniqueness of quasi-Baer and right FI-extending right ring hulls of a semiprime ring is

established.

In Chapter 4 of (Birkenmeier et al., 2009b), they use the concept of boundedly centrally

closed C

-algebras which are somewhat analogous to the centrally closed semiprime rings.

The existence of a bounded centrally closed hull of a C

-algebra is also shown. In particular

all C

-algebras are semiprime and nonsingular, thus every C

-algebra has a quasi-Baer hull.

This shows that every semiprime ring has a quasi-Baer hull.

7

1.4.5 Self-injectivity of rings

G. F. Birkenmeier et al., (2006), use the method of determining the existence and or unique-

ness of right ring hulls of Rin a class