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ON CHAIN DOMAINS, PRIME RINGS and TORSION Preradicals ON CHAIN DOMAINS, PRIME RINGS AND TORSION PRERADICAlS by JOHN ERIC VAN DEN BERG . \ Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics and Applied Mathematics, University of Natal, Pietermari tz burg Pietermaritzburg February, 1995 To my wife Mandy and sons Nicholas and Christopher Preface The work described in this thesis was carried out under the supervision of Dr James Raftery, Department of Mathematics and Applied Mathematics, University of Natal, Pietermaritzburg, from January 1990 to December 1994. The thesis represents original work by the author and has not been submitted in any form to another university. Where use was made of the work of others, it has been duly acknowledged in the text. (i) Acknowledgements I wish to express my deep gratitude to Dr James Raftery for the encouragement and invaluable guidance which he has provided throughout the duration of my PhD studies. I also wish to express my thanks to Dr Raftery for his advice and assistance in preparing the final manuscript, and for the use of his computer. I acknowledge the consideration shown me by my colleagues in the Department of Mathematics and Applied Mathematics, University of Natal Pietermaritzburg, in the allocation of departmental duties during the period of my registration. Thanks are also due to the department for the computing and photocopying facilities which they made available to me. Finally, I would like to thank my wife Mandy for her unfailing support and enormous patience during my studies. (ii) Abstract The main aims of this dissertation are to investigate the notion of an m-Jansian preradical on the module category of a ring, to study rings which are characterized by the behaviour of their m- Jansian torsion preradicals, and to examine (ring and radical theoretic aspects of) a resultant classification of prime rings. To facilitate this, a study of right chain domains is also undertaken. The preliminary chapter (Chapter 0) establishes notational conventions, introduces terminology and presents general algebraic results that will be required in the sequel. Chapter I develops techniques for the construction of noncommutative right chain domains (with a view to application in Chapter II). The main result (Theorem 1.5.1) asserts that every algebraic chain is isomorphic to the chain of all proper (two-sided) ideals of a right chain domain T with identity. (The full ideal lattice may be realized if the requirement that T have identity is dropped.) Furthermore, T may be chosen such that the chains of right ideals strictly between covering pairs in the proper ideal lattice of T contain dually cofinal copies of further preassigned chains; these latter chains must be unbounded above, but may otherwise be quite .arbitrary (Theorem 1.5.3). The ordinal w $ 3, ordered naturally, is an algebraic chain that is not isomorphic to the ideal lattice of any commutative (or even right duo) ring (Proposition 1.5.5). The universal algebraic significance of these results is discussed. Chapter II concerns torsion theoretic aspects of module categories and is pitched at a slightly more general level than the traditional setting. The primitive notion is that of a pre radical on a module subcategory. (A class of (not necessarily unital) modules e over some fixed ring (possibly without identity) is called a module subcategory if e is closed under homomorphic images, submodules, direct products and essential extensions.) The lattice of all torsion preradicals on a module subcategory is modular and algebraic (Proposition III.1.6). Notions of m- Jansian preradicals and m - J ansian topologizing filters (m a regular cardinal) are introduced and studied. The former are preradicals whose pretorsion classes are closed under direct products of fewer than m modules, and the latter are topologizing filters closed under intersections of fewer than m right ideals. A ring R for which every right topologizing filter is m- Jansian is said to be right m-closed. For m ~ Nl every right m- closed ring R satisfies the descending chain condition on (two-sided) ideals (Corollary 11.4.2) and has only finitely many maximal proper ideals (Proposition 11.5.8); also, RfJ(R) is either zero or isomorphic to a finite direct product of finite dimensional matrix rings over division rings (Corollary 11.4.8). If R is a right m- closed ring (m ~ Nl ) then the conditions: (1) RR has Gabriel dimension; (2) J(R) is nilpotent; (3) RR is artinian, are equivalent (Theorem 11.4.15). A ring R (possibly without identity) is artinian if and only if every right topologizing filter on R is closed under arbitrary intersections (Theorem 11.4.16). For every regular cardinal m, there exists a ring which is right m-closed but not right n-closed for any n> m (Theorem 11.4.20). This result applies the (iii) techniques developed in Chapter I. Two rings Rand S (possibly without identity) are said to be Morita * -equivalent if their respective Dorroh Extensions are Morita equivalent. If Rand S are Morita * -equivalent rings and m is a regular cardinal then R is right m- closed if and only if S is right m - closed (Theorem 11.5.10). Consequently, for rings with identity, m- closure is a Morita invariant property (in the classical sense). For a ring R and a regular cardinal m the following conditions are equivalent: (1) For every nonzero a E R there exists a subset X of R such that I X I < m + 1 (here' +' is cardinal addition) and aX has trivial right annihilator; (2) O'(RR) = 0 for all proper m-Jansian torsion preradicals 0' -> T zero on the category of right R-modules (Theorem 11.6.3). (The torsion preradical T zero is defined by Tzero(M) = {x EM: xR = O} for all modules M.) For an arbitrary nonzero cardinal m (possibly finite), a ring R is said to be right prime of bound m if m is the least cardinal for which condition (1) above is satisfied. We use Pr(m) to denote the class of all such rings and define Pr(m) = Uk<m Pr(k). The left analogues of these classes are denoted by P1(m) and P1(m). Several categorical characterizations of the members of Pr (m) are presented for infinite m. Chapter III is devoted primarily to an investigation of the classes P r (m). We study the right bound of primeness of the (row-finite) matrix ring Mm(R) where m is a nonzero cardinal (possibly infinite) and R a prime ring. If D is a division ring then ~(D) E P r (m) if m < No and Mm(D) EPr(m+) if m2:No (Proposition III. 1.4). (Here, m+ denotes the cardinal successor of m.) Using various matrix ring constructions it is proved that the class P r (m) is nonempty for all m> 0 (Proposition III. 1.5). In general, the left bound of primeness of ~(R) (R a prime ring) is not determined simply by n and the bound of primeness of R. For example, if D is a division ring and n is infinite then Mn(D) E PI (k+) where k is the smallest infinite cardinal such that I D I k 2: n (Proposition 111.1.11). We consider the question: for what values of m and n is the class Pr(m)nP1(n) nonempty? The classes Pr (m)nP1(m),Pr (1)nP1(n) and Pr (m)nP1(1) are nonempty for all m,n>O and Pr(m)nP1(n) is nonempty for all m,n>No (Theorem 111.1.13). If m> 1 and R E Pr(m) then R E P1(m) or R E P1(1) (Proposition 111.1.7). The only obstacle to a complete answer to the above question is a longstanding open problem of Goodearl, Handelman and Lawrence: is Pr(NO)~PI(No)UPI(l)? For finite (resp. infinite) m, a ring S with identity is isomorphic to the ring of all linear transformations, written on the right, of an m - dimensional left vector space if and only if S is von Neumann regular, left self-injective with nonzero socle and S E P r (m) (resp. S E P r (m+)) (Proposition III.1.14). For rings with identity and infinite m, membership of Pr (m) is Morita invariant (Theorem III. 1.6). If Rand S are Morita * -equivalent prime rings and m is infinite, then R E Pr(m) if and only if S E Pr(m) (Theorem III. 1. 18). The stipulation that Rand S are prime rings cannot be dropped. Closure properties of the class Pr (m) under various standard overring and subring constructions are investigated. Particular attention is given to semigroup, monoid and polynomial rings as well as monomial algebras. It is known that a right order in a prime ring need not be prime. (iv) If k and m are cardinals with m ~ k, No and R is a right m-order in S (this means that for every subset X of S with I X 1< m, there is a unit u of S, contained in R, such that Xu ~ R) then REP r (k) if and only if S E P r (k) (Theorem III.2.3). If m> 1 and H is an arbitrary semigroup then REP r (m) if the semigroup ring RH is in P r (m) (Proposition 111.3.1). Conversely, for m> 0 we have: (1) if m ~ No, H is a unique product semi group and REP r (m) then RH E P r (m); (2) if H is a cancellative strictly right ordered semigroup and REP r (m) then RH E P r (m); (3) if H is the free product of two nontrivial monoids and REP r (m) then RH E P r (2m) (Proposition 111.3.3).
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