Function ring
A function ring is a special ring of functions in mathematics (more precisely in ring theory ) . These play a major role in abstract algebra , topology , and numerous applications of mathematics in natural sciences.
definition
Be a ring, a non-empty set and
the set of all functions defined on with values in . Then are through
Links are explained that create a ring, the so-called ring of functions.
Important properties
- The ring "inherits" certain properties from , such as commutativity and unity . Other properties, such as freedom from zero divisors , are not "inherited".
- The set of constant functions forms an isomorphic subring of . This can be viewed as a partial ring of .
Examples
- If one chooses as the set of real numbers with the usual addition and multiplication and as an open subset of , one can speak of continuous or differentiable functions. In this case the sets and sub-rings are of . There is a subring of .
Evaluation homomorphism
For a solid is the picture
a ring homomorphism . It is referred to as the evaluation homomorphism or simply the evaluation at the point .
literature
- Albrecht Beutelspacher : Linear Algebra. An introduction to the science of vectors, maps, and matrices. 6th revised and supplemented edition, reprint. Vieweg + Teubner, Wiesbaden 2009, ISBN 978-3-528-56508-4 ( mathematics for first-year students ).
- Gerd Fischer : Textbook of Algebra. Vieweg, Wiesbaden 2008, ISBN 978-3-8348-0226-2 ( Vieweg Mathematics ).