Jacques Lewin

Subrings of Finite Index in Finitely Generated Rings

THEOREM 2. A subring of Jinite index in a finitely generated ring is again finiteZy generated. Both Theorems 1 and 2 are analogs of well-known group-theoretic facts which are easy consequences of the existence of a Schreier set of representatives for a subgroup of a free group (see, e.g., Kurosh, [4). However, another proof of the group theoretic version of Theorem 2 was given by M. Hall [I], and it is his techniques that we apply in part of the proof of our theorem. We also note that once we have Theorem 2, then, again as for groups, we have

On some infinitely presented associative algebras

We prove here that if F is a finitely generated free associative algebra over the field and R is an ideal of F , then F / R 2 is finitely presented if and only if F / R has finite dimension. Amitsur, [1, p. 136] asked whether a finitely generated algebra which is embeddable in matrices over a commutative f algebra is necessarily finitely presented. Let R = F ′, the commutator ideal of F , then [4, theorem 6], F / F ′ 2 is embeddable and thus provides a negative answer to his question. Another such example can be found in Small [6]. We also show that there are uncountably many two generator I algebras which satisfy a polynomial identity yet are not embeddable in any algebra of n x n matrices over a commutative algebra.

Tree Groups and the 4 String Pure Braid Group

Abstract Given a graph Г, undirected, with no loops or multiple edges, we define the graph group on Г, FГ, as the group generated by the vertices of Г, with one relation xy = xy for each pair x and y of adjacent vertices of Г. In this paper we will show that the unpermuted braid group on four strings is an HNN-extension of the graph group Fs, where S = The form of the extension will resolve a conjecture of Tits for the 4-string braid group. We will conclude, by analyzing the subgroup structure of graph groups in the case of trees, that for any tree T on a countable vertex set, Ft is a subgroup of the 4-string braid group. We will also show that this uncountable collection of subgroups of the 4-string braid group is linear, that is, each subgroup embeds in GL(3, R ), as well as embedding in Aut(F), where F is the free group of rank 2.

The growth function of some free products of groups

We give some hypotheses under which it, is easy to compute the growth function of free products and HNN extensions.

The group algebra of a torsion-free one-relator group can be embedded in a field

(* denotes coproduct ( = free product with amalgamation) of groups and rings), it is sufficient that//,, and wGf be linearly disjoint over wH in Gt, i.e., that the multiplication map i/f ®WH wGj —• Gt be injective. If R is a semifir, then [3, Chapter 7] there is a field U(R), the universal field of fractions of R, embedding R such that each automorphism of R extends to an automorphism of U(R). If F is a free group, wF is a semifir and has a universal field of fractions.

On ideals of free associative algebras generated by a single element

In the 1930’s, Wilhelm Magnus [.5], [6] proved some deep theorems about groups presented on a single relation, perhaps the most famous of which are the Freiheitssatz and the solvability of the word problem. With the proper definitions, the statements of Magnus’ theorems translate immediately into conjectures about associative linear algebras. The main object of the present investigation is to show that the Freiheitssatz is indeed true in one-relator algebras of a special type. Specifically, let

On the intersection of augmentation ideals

Let G be a group and ZG its integral group ring. If H is a subgroup of the group 6, the (right) augmentation ideal d,(H) is the right ideal of ZG generated by the elements h 1 with h E H. If HI and Hz are two subgroups of G and H, n H2 = H, then clearly d,(H) C A,(H,) n A&H,). Theorem 1 below characterizes the situation in which A,(H) = A,(H,) n A&H,). As a consequence we obtain a necessary condition for the generalized free product of two finitely generated free groups to be again free and an upper bound for the cohomological dimension of a generalized free product.

The symmetric ring of quotients of a 2-fir

We show that the symmetric ring of quotients of a 2-FIR coincides with the Ore Localization at the set of bounded elements.