KEY WORDS: almost-periodicity, continuous spectrum, Sobolev problems, Coriolis operator, ideal incompressible fluid.Let G be a bounded domain in R s , and let H be the Hilbert space of vector functions L7 = (ul, u2, u3), ui E L2(G), i = 1, 2, 3, with the inner product defined by (U, V) = fG (Ul~l + u2~2 + us~s)dxdydz. 1] for any domain G. It should be pointed out that the qualitative structure of the spectrum of B depends on the configuration of G. On the other hand, the properties of the solutions to the Cauchy problem (1) (such as almost-periodicity in the time variable t) are closely related to the structure of the spectrum. Almost-periodicity of the model problem corresponding to (1) for the case of two space variables, was studied by numerous authors (see the reviews [3,4]). The three-dimensionai problem is much less studied. The structure of the spectrum of the Coriolis operator

1. Introduction. Bass [1] proved that if R is a left perfect ring, then R contains no infinite sets of orthogonal idempotents and every nonzero left R-module has a maximal submodule, and asked if this property characterizes left perfect rings ([1], Remark (ii), p. 470). The fact that this is true for commutative rings was proved by Hamsher [12], and that this is not true in general was demonstrated by examples of Cozzens [7] and Koifman [14]. Hamsher's result for commutative rings has been extended to some noncommutative rings. Call a ring left duo if every left ideal is two-sided; Chandran [5] proved that Bass' conjecture is true for left duo rings. Call a ring R weakly left duo if for every r e R, there exists a natural number n(r) (depending on r) such that the principal left ideal Rr" {r) is two-sided. Recently, Xue [21] proved that Bass' conjecture is still true for weakly left duo rings.In Section 3 of this paper, we first generalize the above results to left quasi-duo rings. We call a ring R a left (right) quasi-duo ring if every maximal left (right) ideal of R is two-sided. It is shown that the class of all weakly left duo rings is properly contained in the class of all left quasi-duo rings but that several basic properties of weakly left duo rings as proved in [22] are valid for left quasi-duo rings. This is the content of Section 2. Furthermore, we establish in Section 4 the equivalence of the following conditions on a left quasi-duo ring R: (1) R is a left P-exchange ring; (2) Throughout this paper all rings are associative with identity and modules are unitary left modules unless otherwise specified. J(R) always denotes the Jacobson radical of a ring R. Homomorphisms of modules will be written on the side of their arguments opposite to scalars.

An associative ring R is said to have stable range 1 if for any a, b e R satisfying aR + bR = R , there exists y e R such that a + by is a unit. The purpose of this note is to prove the following facts. Theorem 3: An exchange ring R has stable range 1 if and only if every regular element of R is unit-regular. Theorem 5: If R is a strongly w-regular ring with the property that all powers of every regular element are regular, then R has stable range 1. The latter generalizes a recent result of Goodearl and Menai [5].

Abstract.An associative ring R is said to have stable range 1 if for any a, b e R satisfying aR + bR = R , there exists y e R such that a + by is a unit. The purpose of this note is to prove the following facts. Theorem 3: An exchange ring R has stable range 1 if and only if every regular element of R is unit-regular. Theorem 5: If R is a strongly w-regular ring with the property that all powers of every regular element are regular, then R has stable range 1. The latter generalizes a recent result of Goodearl and Menai [5].Let R be an associative ring with identity. R is said to have stable range 1 if for any a, b £ R satisfying aR + bR = R, there exists y £ R such that a + by is a unit. This definition is left-right symmetric by Vaserstein [9, Theorem 2]. Furthermore, by a theorem of Kaplansky, all one-sided units are two-sided in rings having stable range 1 (cf. Vaserstein [10, Theorem 2.6]). It is well known that a (von Neumann) regular ring P has stable range 1 if and only if R is unit-regular (see, for example, Goodearl [4, Proposition 4.12]).Call a ring P strongly n-regular if for every element a £ R there exist a number n (depending on a) and an element x £ R such that a" -an+lx. This is in fact a two-sided condition [3]. It is an open question whether all strongly 7r-regular rings have stable range 1. Goodearl and Menai [5] proved that strongly ^-regular rings are unit-regular and, hence, have stable range 1 (Theorem 5.8, p. 278).In this note we first extend the above result for von Neumann regular rings to a larger class of rings, which includes all strongly 7r-regular rings, 7t-regular rings, von Neumann regular rings, and algebraic algebras. As an application of this, we prove that a strongly ^-regular ring P has stable range 1 if powers of every regular element are regular. The latter is a generalization of the abovementioned result of Goodearl and Menai for strongly 7r-regular regular rings. As one can see from our proofs, rings in these classes have a large supply of idempotents.Throughout, R stands for an associative ring with identity and J(R) for the Jacobson radical of R . Modules are unitary right P-modules except otherwise specified. For other undefined terms, readers are referred to [4].Let Mr be a right P-module. Following Crawley and Jonsson [2], Mr is said to have the exchange property if for every module Ar and any two

An associative ring R is said to have stable range one if for any a, b ∈ R satisfying aR + bR = R, there exists y ∈ R such that a + by is right (equivalently, left) invertible. Call a ring R strongly π-regular if for every element a ∈ R there exist a number n (depending on a) and an element x ∈ R such that an = an+1x. It is an open question whether all strongly π-regular rings have stable range one. The purpose of this note is to prove the following Theorem: If R is a strongly π-regular ring with the property that all powers of every nilpotent von Neumann regular element are von Neumann regular in R, then R has stable range one.