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 ${\displaystyle R}$ ${\displaystyle M}$

${\displaystyle \mathbb {F} (M,R):=\{f\colon M\to R\}}$

the set of all functions defined on with values ​​in . Then are through ${\displaystyle M}$ ${\displaystyle R}$

${\displaystyle (f+g)(x):=f(x)+g(x)}$

${\displaystyle (f\cdot g)(x):=f(x)\cdot g(x)}$

Links are explained that create a ring, the so-called ring of functions. ${\displaystyle \mathbb {F} (M,R)}$

Important properties

• The ring "inherits" certain properties from , such as commutativity and unity . Other properties, such as freedom from zero divisors , are not "inherited". ${\displaystyle \mathbb {F} (M,R)}$ ${\displaystyle R}$
• The set of constant functions forms an isomorphic subring of . This can be viewed as a partial ring of . ${\displaystyle R}$ ${\displaystyle F}$ ${\displaystyle R}$ ${\displaystyle F}$

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 . ${\displaystyle R}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle C(M)=\{f\colon M\to R\mid f{\text{ ist stetig}}\}}$ ${\displaystyle D(M)=\{f\colon M\to R\mid f{\text{ ist differenzierbar}}\}}$ ${\displaystyle \mathbb {F} (M,R)}$ ${\displaystyle D(M)}$ ${\displaystyle C(M)}$

Evaluation homomorphism

For a solid is the picture ${\displaystyle a\in M}$

${\displaystyle \phi \colon \mathbb {F} (M,R)\to R}$

${\displaystyle f\mapsto f(a)}$

a ring homomorphism . It is referred to as the evaluation homomorphism or simply the evaluation at the point . ${\displaystyle a\in M}$

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 ).