Pere Menal

Stable range one for rings with many units

Abstract The main purpose of this paper is to prove the stable range 1 condition for a number of classes of rings and algebras. Using a modification of a computation of D.V. Tyukavkin, stable range 1 (and a bit more) is obtained from the following simple condition on a ring R: given any x, y ϵ R, there is a unit u ϵ R such that x − u and y − u-1 are both units. Verification of the latter condition then yields stable range 1 in a number of cases, e.g.: (1) any algebra over an uncountable field, in which all non-zero-divisors are units and there are no uncountable direct sums of nonzero one-sided ideals; (2) any algebra over an uncountable field, with only countably many primitive factor rings, all of which are artinian; (3) the endomorphism ring of any noetherian module over an algebra as in (2); (4) any algebraic algebra over an infinite field; (5) any integral algebra over a commutative ring which modulo its Jacobson radical is algebraic over an infinite field; (6) any von Nuemann regular algebra over an uncountable field, which has a rank function. Using other techniques, it is proved that finite Rickart C∗-algebras, strongly π-regular von Neumann regular rings, and strongly π-regular rings in which every element is a sum of a unit plus a central unit, all have stable range 1. Finally, for an arbitrary commutative ring some overrings with specified stable range properties are constructed, in particular a more or less canonical overring having stable range 1.

On regular rings with stable range 2

In [2] Henriksen proves that if R is a unit regular ring, then every matrix over R admits diagonal reduction. On the other hand it is well known, cf. [l, 4.12; 4.13; 4.151, that a regular ring R is unit regular if and only if R@A 3 R@B implies A 2 B, for all right R-modules A, B. In Theorem 5 below we complete Henriksen’s work [2] by proving that every matrix (possibly rectangular) over a regular ring R admits diagonal reduction if and only if R2@A P ROB implies R@A E B, for all right Rmodules A, B; it is also shown that this holds if and only if R is a regular Hermite ring. As we shall see any regular right Hermite ring is left Hermite, this will follow from the fact, cf. [5], that the stable range of a ring coincides with the stable range of its opposite ring. In Section 1 we also extend some results from unit regular rings to regular rings with finite stable range. In Section 2 we construct some regular rings with stable range 2, thus answering a question of Handelman [l, Problem 491 and a question of Vasershtein [5, Remarks on Theorem 41. G. Bergman, cf. [l, 4.261, constructs a regular ring R such that perspectivity is transitive in the lattice, L(R), of principal right ideals of R, but R is not unit regular. The construction of our examples of rings of stable range 2 was inspired by that example, in fact we offer a regular ring R with stable range 2 such that L(R) EL(S), where S is a subring of R which is unit regular. We see then that R is not unit regular, but it has the same ‘lattice’ properties as a unit regular ring, in particular perspectivity is transitive in L(R). It can be shown, by using the methods we shall develop here, that Bergman’s example has stable range 2, but it appears to us that our examples are simpler than Bergman’s ones. We prove that, unlike the case of unit regular rings, finitely generated projective

On the structure of GL2 over stable range one rings

Abstract Under the first Bass stable range condition on a commutative ring A = 2 A , Costa and Keller have described all normal subgroups of SL 2 A = E 2 A . We generalize this description, dropping the commutativity assumption. Our answer involves quasi-ideals of A rather than ideals (for all commutative rings A = 2 A , every quasi-ideal is an ideal).

On subgroups of GL2 over Banach algebras and von Neumann regular rings which are normalized by elementary matrices

Abstract Let A be an associative ring with 1 and let E 2 A be the group generated by all elementary 2 by 2 matrices over A . In this paper we describe all normal subgroups of E 2 A for all von Neumann regular rings A , as well as for a wide class of rings A containing all Banach algebras. For Banach algebras A , the answer involves “quasi-ideals” of A , which replace ideals of A in the similar results for GL n A , n ⩾ 3, obtained previously by the second author.

On rings whose finitely generated faithful modules are generators

A ring R over which every finitely generated faithful right R-module is a generator of the category mod-R of all right R-modules is called right FPF. FPF ring theory was initiated in this general setting by Faith in order to study from a unified point of view those rings that appear in Morita duality, commutative Prtifer rings and bounded Dedekind prime rings amongst others. The reader can consult Faith and Page’s book [ 141, where most of the basic results on FPF rings are contained. In Section 1 of this paper we construct semiprime FPF rings that are not semihereditary, thus answering [ 14, Question 111. It was an open question [l, p, 173171 whether or not the Pierce stalks of a semiprime FPF ring are FPF. We will answer this in the negative. We close Section 1 by proving that the maximal ring of quotients of a semiprime right FPF ring R satisfying a polynomial identity is the localization of R at the set of all nonzero divisors of the centre Z of R. In the case where R is module finite over Z this was obtained in [l, Proposition 1.91. Section 2 considers centres, Galois subrings, and group rings over FPF rings. By using the methods of Bergman and Cohn, cf. [4, Section 6.21, we show that every integrally closed commutative domain can be realized as the centre of a Bezout FPF domain, and conversely. This answers [14, Question 33 in the negative. We remark that a negative answer for nonsemiprime FPF rings is implicit in [24], where the authors construct a quasi-Frobenius ring whose centre is not quasi-Frobenius. We also provide additional examples showing that Galois subrings of semiprime FPF rings need not be FPF, cf. [14, Question 141. A positive result is that if C is a finite group of automorphisms of a reduced commutative FPF ring, then the fixed ring RG is FPF; this is fairly easy but our examples illustrate and limit the scope for possible generalizations. 425