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A telephone survey of households in 13 states in the southeastern U.S. determined residential fuelwood acquisition and use for heating during the 1991 heating season. Although wood burning accounted for only 10% of the total household heat requirement of the region, it accounted for 51% of the total heat requirements of wood-burning households. One-quarter (25%) of the households burned wood, consuming almost 9 million standard cords. Three-quarters (77%) of wood-burning households reported that wood burning contributed to household heating requirements. Wood as a source of home heating was particularly important to low income households, both rural and urban. Wood is the sole source of home heating for 17% of the wood-burning households in the region.

Throughout, rings are commutative with unit and modules are unital. We prove that R is an elementary divisor ring if and only if every finitely presented module over R is a direct sum of cyclic modules, thus providing a converse to a theorem of Kaplansky and answering a question of Warfield. We show that every Bezout ring with a finite number of minimal prime ideals is Hermite. So, in particular, semilocal Bezout rings are Hermite answering affirmatively a question of Henriksen. We show that every semihereditary Bezout ring is Hermite. Semilocal adequate rings are characterized and a partial converse to a theorem of Henriksen is established.

Manis has developed a valuation theory for commutative rings which extends valuation theory for fields. However his results do not extend the characterization of valuation rings as domains of maximal partial homomorphisms. In this note we show that Manis' theory also generalizes this aspect of valuation theory. Next we show that an overring W of a valuation ring V is not necessarily a valuation ring in any nice sense, but that W is contained in a valuation ring which is a large quotient ring of V. All rings considered will be commutative and have unity element. An element will be called regular if it is not a zero divisor and an ideal will be called regular if it contains a regular element. An overring of a ring R is a ring between R and its total quotient ring. For convenience A\B= {xEAIx(IB}. 1. Maximal partial homomorphisms and valuation pairs. If K is a ring, then by a valuation pair of K, we mean a pair (V, P), where V is a subring of K and P is a prime ideal of V, satisfying any of the followzing equivalent conditions [5, p. 193 ]: (1) If W is a subring of K containing V and if M is a prime ideal of Wwith MnGV=P, then V= W. (2) For all xGK\V, there exists peP such that pxE V\P. (3) There is a mapping v from K onto an ordered abelian group with symbol xo adjoined which satisfies v(xy) = v(x) +v(y) and v(x+y)?min{v(x), v(y)} for all x, yCK and such that V = {xeK|Iv(x) > O} andP= {xe KIv(x) >O0. In [2, Theorem 1 ] the equivalent conditions which define a valuation ring of a field include a condition involving maximal partial homomorphisms. We now establish this result for rings which allow zero divisors. Presented to the Society, August 21, 1970 under the title Valuation pairs and maximal partial homomorphisms; received by the editors June 18, 1970 and, in revised form, August 10, 1970 and November 16, 1970. AMS 1970 subject classifications. Primary 13A15.

In many classical integral domains, given two nonassociates it is possible to find a valuation on the quotient field of the domain which is nonnegative on the domain and for which the nonassociates have different values. Recent work by Griffin, Harrison, and Manis has extended valuation theory to commutative rings with identity which contain zero divisors. In this paper we investigate the separation of nonassociates by valuations for the extended valuation theory. Our main result states that if R is a ring with a von Neumann regular total quotient ring, then nonassociates can be separated by valuations if and only if there is no unit in the integral closure of R which is not a unit in R. All rings considered will be commutative and have identity. A valuation on a ring K is a map v from K onto a totally ordered group with oo adjoined such that v(xy)=v(x)+v(y) and v(x+y)_min{v(x), v(y)} for all x, yeK. The elements of K at which v is nonnegative is a subring denoted by R, and called the valuation ring of v. A subring R of K is a valuation ring with prime ideal P of elements of positive value if and only if for any ring S such that R'S'K and prime ideal M of S with MnR=P we have S=R [3]. We will restrict ourselves to considering valuations on total quotient rings. Let R be a ring, K its total quotient ring. The following definitions will be useful. DEFINITION. A collection fY of valuations on K separates nonassociates of R if given any two nonassociates a, beR, there exists ve17 such that v(a)ov(b). DEFINITION. If R is a subring of a ring S, then R is close to S if every unit of S is also a unit of R. First we restrict ourselves to regular elements. Presented to the Society, September 1, 1971; received by the editors January 29, 1971 and, in revised form, April 26, 1971. AMS 1970 subject classifications. Primary 13A15.

We have constructed the number systems of arithmetic and have studied their properties. In this chapter we look at other “systems” of numbers; these have similar properties but are distinctly different. The porpose of looking at these new systems is to provide “negative instances” for comparison to the system of rational numbers and system of integers. Many elementary school textbooks use these different systems to develop familiarity with and insight into the standard number systems of arithmetic.

This chapter discusses the system of whole numbers and its development and properties, proving either in the exposition or exercises that the properties are valid. Certainly, a child's natural approach to the system of whole numbers is the best for him. When children learn to multiply whole numbers, they know how to add and multiplication is often approached from the standpoint of repeated addition. This approach to multiplication is extremely teachable but has one stumbling block: What does multiplication by zero mean? Is 0 × 3 the sum of zero threes? Numbers allow to make various size comparisons. These comparisons are based on the ordering properties of the whole numbers. This ordering is an important aspect of elementary school arithmetic.

This chapter discusses motion geometry for the definition of parallel lines. When lines are slide images of one another, the lines are called parallel lines. When no slide works, the lines are not parallel. When a pair of parallel lines is cut by a third line, it is a transversal. Any line that crosses two other lines, not necessarily parallel, is called a transversal. When two lines are cut by a transversal, a pair of angles in like positions with respect to the transversal are called corresponding angles. When the lines are parallel, corresponding angles are slide-congruent under a suitable slide along the transversal. The chapter illustrates alternate interior angles, alternate exterior angles, and perpendicularity. It presents a classification of triangles and quadrilaterals in terms of the numbers and kinds of symmetries. A quadrilateral is a four-sided figure. The chapter classifies quadrilaterals into eight types, according to their symmetries.

Publisher Summary This chapter discusses motion geometry for the definition of parallel lines. When lines are slide images of one another, the lines are called parallel lines. When no slide works, the lines are not parallel. When a pair of parallel lines is cut by a third line, it is a transversal. Any line that crosses two other lines, not necessarily parallel, is called a transversal. When two lines are cut by a transversal, a pair of angles in like positions with respect to the transversal are called corresponding angles. When the lines are parallel, corresponding angles are slide-congruent under a suitable slide along the transversal. The chapter illustrates alternate interior angles, alternate exterior angles, and perpendicularity. It presents a classification of triangles and quadrilaterals in terms of the numbers and kinds of symmetries. A quadrilateral is a four-sided figure. The chapter classifies quadrilaterals into eight types, according to their symmetries.

Logic is the language of mathematics. A precise definition of terms and concepts falls under the rubric of logic. This chapter discusses several concepts of elementary logic. The basic building blocks of mathematical statements are propositions. A proposition is a declarative sentence that is either true or false, but not both. The chapter discusses logical connectives that are used to produce new propositions. The letters p and q represent arbitrary propositions. A truth table of a logical proposition shows the conditions under which the logical proposition is true (T) and those under which it is false (F). Truth tables are used as a convenient device to present the reader with an intuitive feeling for the various connectives.

The concept of function pervades most of mathematics. Its power Hes in the fact that it unifies many ideas in mathematics from the most elementary to the most advanced. The notions of number and binary operation are examples of functions used in the elementary classroom. In almost every branch of mathematics current research deals with functions in some way. The fact that the concept of function is so important makes a clear understanding of functions and a facility with functional notation a must for any serious student or teacher of mathematics.

One of the first topics in arithmetic in the elementary school is counting. The television program Sesame Street spends hours preparing preschool children for counting. Counting and numbers are certainly basic to all of arithmetic. The technique of counting in the first grade involves a one-to-one correspondence between objects and an ordered set called the counting numbers (whole numbers). The ease of this counting depends on the niceness or concreteness of the set. Given a set of objects that can be manipulated, counting them is indeed trivial and dull. However, counting a set that is described by certain properties and structures can be complicated and exciting. This chapter discusses some techniques of counting these sets. The number of ways of making several decisions in succession is the product of the number of choices that can be made in each decision. If, in an arrangement of K objects selected from a set of n different objects, the order of arrangement is important, so that different arrangements of the same set of K objects are regarded as different, then each such arrangement is called a permutation.

This chapter discusses systems of rational numbers and real numbers. It discusses addition, multiplication, ordering, and division of rational numbers. There are an infinite number of rational numbers between any two rational numbers. The set of all real numbers that are not rational is called the set of irrational numbers. It can be shown that between any two irrational numbers, there are an infinite number of irrational numbers. Because there are an infinite number of rational numbers between any two rational numbers, it appeared as though the rational numbers are rather tightly packed on the number line. However, there is a fundamental difference between the rational numbers and the real numbers. The real number line is continuous, whereas the rational number line is not continuous. The development of calculus is highly dependent on this special property of the real numbers. Any two rational numbers have a product and their product is a rational number as the denominators' being nonzero guarantees that the product has nonzero denominator.

This chapter discusses certain aspects of decimal numeration system and explains how this system and the structural properties of mathematics blend together in the formation of the conventional algorithms for computation. A numeration system is an organized, logical scheme for naming numbers. The system commonly used by the people in most countries is based on the number ten and, hence, is called a decimal system. The system uses 10 numerals called digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Any whole number can be designated by these digits used alone or in conjunction with other digits. This highly efficient system incorporates three powerful concepts; addition, multiplication, and place value. The form of expressing a number as the sum of multiples of powers of 10 is called expanded notation.

Publisher Summary The sets of numbers used in arithmetic are important as the operations, namely, addition, subtraction, and multiplication can be used to combine numbers to get other numbers. Such operations are called binary operations. The study of arithmetic by children in elementary school is for a large part the study of the properties of the binary operations addition and multiplication on the various systems of numbers they encounter. This chapter describes the notion of binary operation to provide teachers with an understanding of and an appreciation for the complexity of the concept. A binary operation on a set may have additional properties. There are two in particular that are of interest in arithmetic: commutativity and associativity.