Abstract
An algebraic structure is a set with operations between its elements that follow certain rules. As an example of such a structure consider the integers and the operation ‘+.’ What are the properties of this addition? Already in elementary school one learns that the sum a + b of two integers a and b is another integer. Moreover, there is a number 0 such that 0 + a = a for every integer a, and for every integer a there exists an integer \(-a\) such that \((-a)\) + a = 0. The analysis of the properties of such concrete examples leads to definitions of abstract concepts that are built on a few simple axioms. For the integers and the operation addition, this leads to the algebraic structure of a group.
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Notes
- 1.
“... die mathematische Methode hingegen schreitet von den einfachsten Begriffen zu den zusammengesetzteren fort, and gewinnt so durch Verknüpfung des Besonderen neue and allgemeinere Begriffe.”
- 2.
Named after Niels Henrik Abel (1802–1829), the founder of group theory .
- 3.
The concept of zero divisors was introduced in 1883 by Karl Theodor Wilhelm Weierstraß (1815–1897).
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© 2015 Springer International Publishing Switzerland
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Liesen, J., Mehrmann, V. (2015). Algebraic Structures. In: Linear Algebra. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-24346-7_3
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DOI: https://doi.org/10.1007/978-3-319-24346-7_3
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