Free Group Algebras in Division Rings with Valuation II

We apply the filtered and graded methods developed in earlier works to find (noncommutative) free group algebras in division rings.If $L$ is a Lie algebra, we denote by $U(L)$ its universal enveloping algebra. P. M. Cohn constructed a division ring $\mathfrak{D}_{L}$ that contains $U(L)$. We denote by $\mathfrak{D}(L)$ the division subring of $\mathfrak{D}_{L}$ generated by $U(L)$.Let $k$ be a field of characteristic zero, and let $L$ be a nonabelian Lie $k$-algebra. If either $L$ is residually nilpotent or $U(L)$ is an Ore domain, we show that $\mathfrak{D}(L)$ contains (noncommutative) free group algebras. In those same cases, if $L$ is equipped with an involution, we are able to prove that the free group algebra in $\mathfrak{D}(L)$ can be chosen generated by symmetric elements in most cases.Let $G$ be a nonabelian residually torsion-free nilpotent group, and let $k(G)$ be the division subring of the Malcev–Neumann series ring generated by the group algebra $k[G]$. If $G$ is equipped with an involution, we show that $k(G)$ contains a (noncommutative) free group algebra generated by symmetric elements.

Free symmetric algebras in division rings generated by enveloping algebras of Lie algebras

For any Lie algebra L over a field, its universal enveloping algebra U(L) can be embedded in a division ring 𝔇(L) constructed by Lichtman. If U(L) is an Ore domain, 𝔇(L) coincides with its ring of fractions. It is well known that the principal involution of L, x ↦ -x, can be extended to an involution of U(L), and Cimpric proved that this involution can be extended to one on 𝔇(L). For a large class of noncommutative Lie algebras L over a field of characteristic zero, we show that 𝔇(L) contains noncommutative free algebras generated by symmetric elements with respect to (the extension of) the principal involution. This class contains all noncommutative Lie algebras such that U(L) is an Ore domain.

On Skew Linear Groups Associated with Certain Soluble-by-Finite Groups

Let O1 denote the class of groups G such that every group ring of G (over a field) is an Ore domain. Several approaches to the correction of the proof of a result concerning subrings generated by certain O1-groups are given.

Isomorphisms of Prime Goldie Semi-Principal Left Ideal Rings, II

A prime Goldie ring K, in which each finitely generated left ideal is principal is the endomorphism ring E(F, A) of a free module A, of finite rank, over an Ore domain F. We determine necessary and sufficient conditions to insure that whenever K ≅ E(F, A) ≅ E(G, B) (with A and B free and finitely generated over domains F and G) then (F, A) is semi-linearly isomorphic to (G, B). We also show, by example, that it is possible for K ≅ E(F, A ) ≅ E(G, B), with F and G, not isomorphic.

When rings of differential operators are maximal orders

Let A be a commutative domain, finitely generated as an algebra over a field k of characteristic zero and write (A) for the ring of k -linear differential operators. Then A is an Ore domain with quotient division ring, say Q. Our main result is that A is a maximal order in Q if and only if (i) A = ∩{Ap: height (p) = 1} and (ii) A is geometrically unibranched. In this case A is also a Krull domain with no reflexive ideals. We also determine some conditions under which A is simple.

Simple Type III Self-Injective Rings and Rings of Column-Finite Matrices

Relatively little is known about simple, Type III, right self-injective rings Q. This is despite their common occurrence, for example as Qmax(R) for any prime, nonsingular, countable-dimensional algebra R without uniform right ideals. (In particular Q can be constructed with a given field as its centre.) As with their directly finite, SP(1), right self-injective counterparts, division rings, there are few obvious invariants apart from the centre.One reason perhaps why little interest has been shown in their structure is that the usual construction of such Q, namely as a suitable Qmax(R), is not concrete enough; in general R sits far too loosely inside Q and not enough information transfers to Q from R. Thus, for example, taking R to be a non-right-Ore domain and Q = Qmax(R) tells us little about Q (although it has been conjectured that all Q arise this way).

Rings Characterized by their Cyclic Modules

A ring R (with identity element) is called a right PCI-ring if and only if every proper cyclic right R-module is injective; that is, if C is a cyclic right R-module then either C ≌ R or C is injective. Faith [3, Theorems 14 and 17] (or see [2, Proposition 6.12 and Theorem 6.17]) proved that if a ring R is a right PCI-ring then R is semiprime Artinian or R is a simple right semihereditary right Ore domain. These latter rings we shall call simple rightPCI-domains. Examples of non-Artinian simple right PCI-domains were produced by Cozzens [1]. The object of this paper is to examine rings with similar properties and thus extend Faith's results.

Lattice-Ordered Rings of Quotients

R. E. Johnson (10), Utumi (18), and Findlayand Lambek (7) have defined for each ring R a unique maximal "ring of right quotients" Q. When R is a commutative integral domain (in this paper an integral domain need not be commutative) or an Ore domain, then Q is the usual division ring of quotients of R. Moreover, it is well known that in these special cases, if R is totally ordered, then so is Q.The main purpose of this paper is to study the ring of quotients Q, and in particular its order properties, for certain lattice-ordered rings R.