Right orders in full linear rings (1971)
AuthorsO'Meara, K. C.show all
The thesis is a study of right orders in a (left) full linear ring Q = HomD(V,V), V a right vector space over a division ring D. Chapter I, as well as summarizing known results needed in the sequel, outlines the method of attack. In particular, it justifies the hypothesis “suppose Q is a (Johnson) right quotient ring of R” which appears frequently in subsequent chapters. One of the principal results in Chapter II is that right orders in full linear rings of countable dimension must be prime rings whereas in the uncountable case this need not be so. In fact, results in Chapter II suggest that a complete description of a right order in Q may be rather difficult if dim QQ is uncountable (here dim MR refers to the uniform dimension of a module MR). Chapter III is a study of intrinsic extensions of prime rings. This study is required by the condition that regular elements of a right order R in Q be units in Q, since this actually implies Q is left intrinsic over R if Q has infinite dimension. The principal result on intrinsic extensions says that if S is a prime ring with zero right singular ideal, but not an integral domain, and if S contains uniform right ideals then S is a right quotient ring of any prime ring over which it is right intrinsic. This result has several interesting corollaries for example, if dim QQ is countable then a right order R in Q must have Q also as a left quotient ring. The main goal of Chapter IV is finding suitable conditions to ensure a ring R will have a full linear ring as its left-flat epimorphic hull. To this end, two conditions are introduced: condition (A) which requires closed right ideals of R to be essential extensions of finitely generated right ideals, and the existence of a “reducing pair” or elements. The latter means a pair (β,γ ) for which βR, γR and β^r; + γ^r are large right ideals of R. Taken together, these two conditions on a ring R having Q as a right quotient ring imply that for each xϵQ there exists c⟎R such that c has a right inverse in Q and xc⟎R. Ample evidence is produced to show that reducing pairs for infinite dimensional rings R play a similar role to primeness for finite dimensional R, in so far as determining when R is a right order in Q. An earlier result of Chapter IV says that if R is a prime ring then R is a right order in Q only if R+ socle Q is, that is, in so far as a study of right oders in Q is concerned, we can suppose R contains the socle of Q. The final chapter contains, among other things, two internal characterizations of a right order R n a infinite dimensional full linear ring. One of these says: A ring R is a right order in a left full linear ring of right dimension א = א ₀ if and only if the following conditions are satisfied. (i) R is a (Johnson) irreducible ring containing uniform right ideals and dim RR = א. (ii) The closed right ideals of R are right annihilator ideals and if B is a right annihilator ideal with dim BR = dim RR then B has the form b^ℓr for some b ⟎ R with b^r = 0. (iii) R possesses a reducing pair of elements.