Advertisement

# Basics of Ring Theory

• Joseph L. Awange
• Béla Paláncz
Chapter

## Abstract

This chapter presents the concepts of ring theory from a geodetic and geoinformatics perspective. The presentation is such that the mathematical formulations are augmented with examples from the two fields. Ring theory forms the basis upon which polynomial rings operate. As we shall see later, exact solution of algebraic nonlinear systems of equations are pinned to the operations on polynomial rings. In Chap. , polynomials will be discussed in detail. In order to understand the concept of polynomial rings, one needs first to be familiar with the basics of ring theory. This chapter is therefore a preparation for the understanding of the polynomial rings presented in Chap. . Ring of numbers which is presented in Sect. 2.2 plays a significant role in daily operations. They permit operations addition, subtraction, multiplication and division of numbers. For those engaged in data collection, ring of numbers play the following role;
• they specify the number of sets of observations to be collected,

• they specify the number of observations or measurements per set,

• they enable manipulation of these measurements to determine the unknown parameters.

## Keywords

Natural Number Geographical Information System Rational Number Polynomial Ring Ring Theory
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 70.
Becker T, Weispfenning V (1993) Gröbner bases. A computational approach to commutative algebra. Graduate text in mathematics, vol 141. Springer, New YorkGoogle Scholar
2. 71.
Becker T, Weispfenning V (1998) Gröbner bases. A computational approach to commutative algebra. Graduate text in mathematics, vol 141, 2nd edn. Springer, New YorkGoogle Scholar
3. 229.
Grafarend EW, Lohse P, Schaffrin B (1989) Dreidimensionaler Rückwärtsschnitt. Zeitschrift für Vermessungswesen 114:61–67, 127–137, 172–175, 225–234, 278–287Google Scholar
4. 284.
Irving RS (2004) Integers, polynomials, and rings. Springer, New YorkGoogle Scholar
5. 319.
Lam TY (2003) Exercises in classical ring theory. Springer, New York/TokyoGoogle Scholar
6. 366.
McCoy NH, Janusz GJ (2001) Introduction to abstract algebra. Harcout Academic, San DiegoGoogle Scholar
7. 372.
Monhor D (2001) The real linear algebra and linear programming. Müszaki Könyvkiadó, BudapestGoogle Scholar
8. 465.
Shut GH (1958/1959) Construction of orthogonal matrices and their application in analytical Photogrammetrie. Photogrammetria XV:149–162Google Scholar
9. 474.
Stillwell J (2003) Elements of number theory. Springer, New York
10. 485.
Thompson EH (1959) A method for the construction of orthogonal matrices. Photogrammetria III:55–59Google Scholar
11. 486.
Thompson EH (1959) An exact linear solution of the absolute orientation. Photogrammetria XV:163–179Google Scholar
12. 490.
Trefethen LN, Bau D (1997) Numerical linear algebra. SIAM, Philadelphia
13. 549.
Zhang S (1994) Anwendung der Drehmatrix in Hamilton normierten Quaternionenen bei der Bündelblock Ausgleichung. Zeitschrift für Vermessungswesen 119:203–211Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 2016

## Authors and Affiliations

• Joseph L. Awange
• 1
• Béla Paláncz
• 2
1. 1.Curtin UniversityPerthAustralia
2. 2.Budapest University of Technology and EconomicsBudapestHungary