A ring R for which every finitely generated right submodule of SR, the left flat epimorphic hull of R, is projective is termed an extended semi-hereditary ring. It is shown that several of the characterizing properties of Prufer domains may be generalized to give characterizations of extended semi-hereditary rings. A suitable class of PP rings is introduced which in this case serves as a generalization of commutative integral domains.

An elementary and self-contained account of analytic Jordan decomposition of matrix-valued analytic functions is given. An integral representation for their eigenvalues is obtained. This leads to estimates of the differences in eigenvalues and the number of points of degeneracy.

We present a direct combinatorial proof of the characterization of the degree of transivity of a finite permutation group in terms of the Bell numbers.

Subject classification (Amer. Math.Soc. (MOS) 1970): 20 B 20.

Groups with the property of the title were considered by Chillag (1977); this paper completes his results by showing that, with known exceptions, they are triply transitive.

If A is a T-orderable R-module and S is a multiplicative subsemigroup of R, each s∈S acting as a monomorphism of A, then it is possible sometimes A in a T-orderable R-module on which each s∈S as an automorphism. We show that such an embedding does not always exist and, by generalizing a theorem of Kokorin and Kopytov, provide a partial answer to the question “when is such an embedding possible?”

Subject classification (Amer. Math. Soc. (MOS) 1970): 06 A 70, 16 A 08, 16 A 64.

We investigate conditions on a permutation group G sufficient to ensure that G fixes a graph in any switching class of graphs that it stabilizes. Our main result gives a necessary and sufficient condition for a dihedral group G to have this property.

Ore (1942) studied the automorphisms of finite monomial groups and Holmes (1956, pp. 23–93) has given the form of the automorphisms of the restricted monomial groups in the infinite case. The automorphism group of a standard wreath product has been studied by Houghton (1962) and Segal (1973, Chapter 4). Monomial groups and standard wreath products are both special cases of permutational wreath product. Here we investigate the automorphisms of the permutational wreath product and consider to what extent the results holding in the special cases remain true for the general construction. Our results extend those of Bunt (1968).

We show that rings for which every non-constant multiplicative endomorphism is additive are trivial or power rings (that is, rings R such that R = R2 and x2 = 0 = x+x for all x ∈ R) and that if R is a power ring for which every multiplicative endomorphism is additive, then End (R) is a zero semigroup or a semilattice according to how the product is defined.

In the last few years, the oscillatory behavior of functional differential equations has been investigated by many authors. But much less is known about the first-order functional differential equations. Recently, Tomaras (1975b) considered the functional differential equation and gave very interesting results on this problem, namely the sufficient conditions for its solutions to oscillate. The purpose of this paper is to extend and improve them, by examining the more general functional differential equation

The divisor density of an integer sequence is defined: this measures the tendency of divisors of (almost all) integers to belong to the sequence. A proof of a conjecture of Erdös is given and this is linked to some previous conjectures of the author's concerning distribution (mod 1) of functions of the divisors.

For n≥2, a hypersurface in the open unit ball Bn in is constructed which satisfies the generalized Blaschke condition and is a uniqueness set for all Hp(Bn) with p>0. If n≥3, the hypersurface can be chosen to have finite area.

Subject classification (Amer. Math. Soc. (MOS) 1970): primary 32 A 10.

This paper is devoted to a study of pseudocomplements in groupoids. A characterization of an intraregular groupoid is obtained in terms of prime ideals. It is proved that the set of dense elements of an intraregular groupoid S with 0 is the intersection of all the maximal filters of S and that the set of normal elements of an intraregular groupoid closed for pseudocomplements forms a Boolean algebra under natural operations. It is shown that the pseudocomplement of an ideal of an intraregular groupoid with 0 is the intersection of all the minimal prime ideas not containing it.

A generalized version of the Knaster-Kuratowski-Mazurkiewicz theorem is obtained and used to generalize Ky Fan's minimax principle. This result is applied to a variational inequality.

In Russell (1973) a Riemann-type necessary and sufficient condition was given for the existence of (defined also in Russell (1975)) when f was bounded and g was k-convex in [a′, b′]. In this paper we present necessary and sufficient conditions for the existence of a particular Stieltjes-type integral without imposing a convexity condition upon g. These conditions are used to obtain an additivity result for the integral over adjoining intervals without any additional restrictions being imposed upon the functions involved.

The consequences of the law (Ga(1), hellip, Ga(n)) = 1 in a basic p-group are examined. The principal tools are a combinatorial analysis of the lattice of n-tuples of positive integers and a theorem of the author about higher-commutator subgroups.

Subject classification (Amer. Math. Soc. (MOS) 1970): 20 D 15.

As dual to the notion of “finitely injective modules” introduced and studied by Ramamurth and Rangaswamy (1973), we define a right R-module M to be finitely projective if it is projective. with respect to short exact sequences of right R-modules of the form 0 → A → B → C → 0 with C finitely generated. We have completely characterized finitely projective modules over a Dedekind domain. If R is a Dedekind domain, then an R-module M is finitely projective if and only if its reduced part is torsionless and coseparable.

For a Dedekind domain R, finite projectivity, unlike projectivity is not hereditary. But it is proved to be pure hereditary, that is, every pure submodule of a finitely projective R-module is finitely projective.

This paper is a contribution to the problem of characterizing the ℵ0-categorical Stone algebras. If the dense set is either finite or a chain, the problem is solved by reducing it to the ℵ0-categoricity of the skeleton and the dense set, solutions for these being known. If the dense set is a Boolean algebra, we show that this type of reduction works for certain subclasses but not for all such algebras. For generalized Post algebras the characterization problem is solved completely.

The integral structure of a simple Lie algebra L of Chevalley type over a field F of fractions of an integral domain D is studied. Sandwich relations for sufficiently large orders are obtained, including a new general sandwich relation for orders of L in case D is an integrally closed Noetherian domain. Generalizations of the principal results of Hyman (1966) in the case when D is a ring of algebraic integers are obtained, using techniques developed by the author and Stewart (1973) which are applied to certain orders in L that arise in a natural fashion from the Chevalley basis.

Subject classification (Amer. Math. Soc. (MOS) 1970): primary 17 B 20; secondary 17 B 10, 17 B 45, 20 G 15, 20 G 05.

Let T be a continuous t-norm (a suitable binary operation on[0, 1]) and Δ + the space of distribution functions which are concertratede on [0,∞. theτT product of any F, G in Δ+ is defined at any real x by , and the pair (Δ+, τT) forms a semigroup. Thus, given a sequence {Fi} in Δ+, the n-fold product τT(F1 … Fn) is well-defined for each n. Moreover, that resulting sequence {τT(F1, …, Fn)} is pointwise non-increasing and hence has a weak limit. This paper establishes a convergence theorem which yields a representation for this weak limit. In addition, we prove the Zero-One law that, for Archimedean t-norms, the weak limit is either identically zero or has supremum 1.