Erol Barbut

A generalization of rotational tournaments

Abstract The notions of rotational tournament and the associated symbol set are generalized to r -tournaments. It is shown that a necessary and sufficient condition for the existence of a rotational r -tournament on n vertices is ( n , r )=1. A scheme to generate rotational r - tournaments is given, along with some examples.

Torsion theories over commutative rings

Abstract The definition of an h-local integral domain is generalized to commutative rings. This new definition is in terms of Gabriel topologies; i.e., in terms of hereditary torsion theories over commutative rings. Such commutative rings are characterized by the decomposition of torsion modules relative to a given torsion theory. Min-local commutative rings constitute a special case. If R is a min-local commutative ring, then an injective cogenerator of a nonminimal Gabriel topology of R is the direct product of the injective cogenerators of all the localizations of the given Gabriel topology. The ring of quotients of a min-local commutative ring with respect to a nonminimal Gabriel topology can be canonically embedded into the product of rings of quotients of localizations. All the Gabriel topologies of commutative valuation rings and their rings of quotients are described. If R is a min-local Prufer commutative ring, then the ring of quotients of R with respect to any nonminimal Gabriel topology of R can be canonically embedded into a product of rings of quotients of localizations, each of which is a valuation ring or a topological completion of a valuation ring.

Rings that are FGC relative to filters of ideals

All our rings will be commutative with identity not equal to zero. Also R will always denote a ring. is a filter of ideals of R if is a nonempty set of ideals of R satisfying: I ∈ and J is an ideal of R with I ⊂ J , then J ∈ , and if I , J ∈ then I ∩ J ∈ . A Gabriel topology of R is a filter of ideals of R satisfying: if J ∈ and I is an ideal of R with ( I : x )∈ for all x ∈ J , then I ∈ . See the B. Stenstrom text [6]. We say that a ring R is an FGC ring if every finitely generated R -module is a direct sum of cyclic R -modules. Use mspec R for the set of all maximal ideals of R .

Decomposing torsion modules

E. Matlis proved that if R is an integral domain with quotient field Q and K is the R-module Q/R, then all torsion R-modules decompose into a direct sum of local submodules if and only if K decomposes into a direct sum of local submodules. Thus K is a test module to determine whether torsion modules decompose. We generalize this result to commutative rings. If R is a commutative ring and a torsion theory of R is given by a Gabriel topology , then form the ring of quotients R and let K be the cokernel of the canonical ring homomorphism from R to R. In some special cases, every -torsion R-module decomposes into a direct sum of local submodules if and only if K decomposes. However, there is an example where this is not the case. The principal result is: given R,  and K, there is a related filter K of ideals of R, which is a subset of , such that all K-pretorsion R-modules decompose into a direct sum of local submodules if and only if K decomposes. The relationship between  and K is investigated.

Classroom notes An interesting property of the parabola

In [1], the following property of parabolas is noted: For any interval of fixed length, a, the area enclosed by the secant line bounded by the end‐points of the interval and the parabola given by y = ax2 + bx + cremains constant, and is equal to |a|α/6. The author proves this by calculus, and asks whether there is an intuitively obvious’ heuristic argument that proves it. In this note I would like to offer a geometrically convincing heuristic argument based on the mean value theorem (MVT). The result can then be verified using linear algebra.

Decomposing finitely generated torsion modules

Let R be a commutative ring and let F be a filter of ideals of R. We give three characterizations of the rings with the property that all finitely generated F pretorsion R modules decompose into a direct sum of cyclic R submodules. One of these characterizations involves decomposing the filter F into a product of subfilters. This filter decomposition is shown to be unique, and is used to characterize several other ring properties relative to the filter of ideals.

Abstract algebra for high school teachers: an experiment with learning groups

We consider a different classroom approach to teaching abstract algebra for prospective high school teachers by breaking the class up into small groups, providing them with worksheets which organize the material into problem sets that can be solved by these groups. We report on the result of this approach in a particular classroom setting.

Probabilistic aspects of some problems in combinatorial geometry

Let Q denote the unit square, [0,1] X [0,1]. Problem 1: Given two random points in Q what is the probability that the line determined by these two points will intersect two adjacent sides of Q? Problem 2: Given four random points in Q what is the probability that the quadrilateral determined by these four points is convex? A theoretical answer to the first problem is obtained and confirmed by computer simulation. An answer to the second problem is obtained by computer simulation.

Meandering around the mean value theorem

Often the mean value theorem (MVT) is stated and proved in introductory calculus courses. One example that is frequently used is a quadratic polynomial, where the slope of the secant between two points on the graph of the function is equal to the slope of the tangent at the midpoint of the abcissas of the two points. It can be easily seen that a polynomial function is a quadratic if, and only if, the point whose existence is asserted by the MVT is unique and lies at the midpoint of the interval. We ask whether there are other functions for which some uniquely‐chosen relevant point for the tangent is a constant fraction of the distance between two abcissas. Our discussion follows one possible evolution of this question. We give an answer for polynomial functions of degree n, and state a more general result in terms of the Taylor series expansion. We conclude by giving some examples and conjectures for further extensions. It is our hope that this could be used to illustrate the process of mathematical disco...