We call a ring R (associative and with identity) for which every quasi-injective right R-module is injective a QII-ring. Similarly R is called an SSI-ring when every semisimple right R-module is injective. Clearly every semisimple, artinian ring is a QII-ring and every QII-ring is an SSI-ring. One then asks whether these inclusions among classes of rings are proper. The purpose of this note is to point out an instance when SSI implies QII It is then easy to see that an example of Cozzens shows that the class of QII-rings properly contains the class of semisimple, artinian rings.

En étudiant certains cônes de fonctionnelles affines positives, nous avons été amenés à considérer les polynômes qui sont positifs sur l'intervalle [-1, 1]. En fait, la fonctionnelle affine

The Banach contraction principle has been generalized by Tan [6] to the mappings in separated locally convex spaces. We show that the result of Sehgal [5] and also of Holmes [3] can be generalized in the same way.

Throughout this note, we let X be a separated locally convex space, U a base for the closed absolutely convex neighborhoods of the origin O in X, K a nonempty subset of X, and Ta mapping from K to K.

Let A be an algebra with 1 over a field F and let B be a fixed F-basis of A. Let F〈x〉=F〈x1,…, xn,…,〉 be the free algebra over F in noncommutative indeterminates x1,…, xn,…, and denote by A F 〈x〉 the free product of A and F〈x〉 over F. The elements of A F 〈x〉 of the form varies, repetitions allowed) form an F-basis of A F 〈x〉. They will be referred to as basis monomials, and the involved in a particular basis monomial will be called the coefficients of that basis monomial.

C(X) (resp. C *(X)) will denote as usual the ring of all (resp. all bounded) continuous functions into the real line R. Define C #(X) to consist of all f∊C(X) whose image M(f) in the residue class ring C(X)jM is real for every maximal ideal M in C(X). Then C # shares with C * the property of being an intrinsically determined subalgebra of C. Then C* shares with C* the property of being an intrinsically determined subalgebra of C. The compactification corresponding to C# (as uniformity determining subalgebra of C*) is thus also an intrinsically determined one. We show that this compactification is well known and "natural" in the cases of several elementary spaces X.

The classical uniqueness theorem of Nagumo [1] for ordinary differential equations is as follows.

Theorem. If f(t, y) is continuous on 0≤t≤1, -∞<y<∞ and if

then there is at most one solution to the initial value problem y'=f(t, y), y(0)=0.