Theorem about the establishment homomorphism

The theorem of insertion homomorphism is an important theorem from ring theory , which allows to insert into the polynomials in the sense of abstract algebra instead of other objects (elements of a ring expansion ). ${\displaystyle X}$

Formulation of the sentence

The statement of the sentence is

Let be a commutative , unitary (i.e. with one element 1) ring , the polynomial ring above and a ring expansion. Then for each element is the figure ${\displaystyle R}$ ${\displaystyle R[X]}$ ${\displaystyle R}$ ${\displaystyle R\subseteq S}$ ${\displaystyle r\in S}$

${\displaystyle F_{r}\colon R[X]\rightarrow S}$

${\displaystyle \sum _{i\in \mathbb {N} _{0}}{a_{i}X^{i}}\mapsto \sum _{i\in \mathbb {N} _{0}}{a_{i}r^{i}}}$

a homomorphism of rings . It is referred to as the insertion homomorphism to . ${\displaystyle F_{r}}$ ${\displaystyle r}$

For and you write short instead of . With this notation the homomorphic properties read and for all . ${\displaystyle r\in S}$ ${\displaystyle f\in R[X]}$ ${\displaystyle F_{r}(f)}$ ${\displaystyle f(r)}$ ${\displaystyle (f+g)(r)=f(r)+g(r)}$ ${\displaystyle (f\cdot g)(r)=f(r)\cdot g(r)}$ ${\displaystyle f,g\in R[X]}$

The homomorphism properties of is easy to check. The ring must therefore be unitary, because then an element of is and each polynomial can thus be clearly represented in the form with for almost all . ${\displaystyle F_{r}}$ ${\displaystyle R}$ ${\displaystyle X:=(0,1,0,0,\dotsc )}$ ${\displaystyle R[X]}$ ${\displaystyle f\in R[X]}$ ${\displaystyle \textstyle f=\sum _{i\in \mathbb {N} _{0}}{a_{i}X^{i}}}$ ${\displaystyle a_{i}=0}$ ${\displaystyle i\in \mathbb {N} _{0}}$

One can dispense with the requirement that there is commutative. It suffices to assume that with all elements from is interchangeable. ${\displaystyle R}$ ${\displaystyle r\in S}$ ${\displaystyle R}$

meaning

In the sense of abstract algebra , polynomials are not functions , as in analysis , but (infinite) sequences of ring elements and not unknowns, but the concrete sequence . However, the theorem of insertion homomorphism shows how one can insert different objects instead of different objects in algebra . The polynomial serves as a "pattern" for the formation of . ${\displaystyle X}$ ${\displaystyle X=(0,1,0,0,...)}$ ${\displaystyle X}$ ${\displaystyle f\in R[X]}$ ${\displaystyle f(r)}$

This is illustrated by the following example.

Let be the polynomial over the field of real numbers and be a (2x2) matrix with real entries . This is an element of the matrix ring that can be understood as a ring expansion of the field of real numbers (because the real numbers are isomorphic to the ring of the matrices of the form with , which is a sub-ring of the matrix ring ). So we can calculate: ${\displaystyle f}$ ${\displaystyle x^{2}+x-6}$ ${\displaystyle M}$ ${\displaystyle M=\left[{\begin{smallmatrix}2&1\\0&3\end{smallmatrix}}\right]}$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{2\times 2}}$ ${\displaystyle \left[{\begin{smallmatrix}a&0\\0&a\end{smallmatrix}}\right]}$ ${\displaystyle a\in \mathbb {R} }$ ${\displaystyle \mathbb {R} ^{2\times 2}}$ ${\displaystyle f(M)}$

${\displaystyle f(M)={\begin{bmatrix}2&1\\0&3\end{bmatrix}}\cdot {\begin{bmatrix}2&1\\0&3\end{bmatrix}}+{\begin{bmatrix}2&1\\0&3\end{bmatrix}}-{\begin{bmatrix}6&0\\0&6\end{bmatrix}}={\begin{bmatrix}0&6\\0&6\end{bmatrix}}}$

Historical outlook

The whole of modern algebra has emerged from the study of algebraic equations , for example the type where stands for the unknown quantity and the coefficients come from a field or, more generally, from a ring . Such an equation is called polynomial . If you want to solve it, you usually look at the associated polynomial function , which assigns the function value to an element , and tries to determine its zeros. Strictly speaking, you also have to define the domain within which you can vary. This can be itself, or for real or complex numbers (more generally a field or ring expansion of the coefficient range). ${\displaystyle \textstyle a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}=\sum _{i=0}^{n}a_{i}x^{i}=0}$ ${\displaystyle x}$ ${\displaystyle a_{1},...,a_{n}}$ ${\displaystyle K}$ ${\displaystyle R}$ ${\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}}$ ${\displaystyle x}$ ${\displaystyle f(x)}$ ${\displaystyle x}$ ${\displaystyle K}$ ${\displaystyle K=\mathbb {Q} }$

One problem here is to find a suitable definition range that contains "all" zeros as far as possible. Another problem arises if one would like to consider a finite body with elements . Then there is, for example, a polynomial function that vanishes entirely even though its coefficients are not all zero. It follows from this that, depending on the domain of definition of the polynomial function assigned to the algebraic equation, one cannot necessarily infer the coefficients of this equation. ${\displaystyle x}$ ${\displaystyle K}$ ${\displaystyle a_{1},...,a_{n}}$ ${\displaystyle \textstyle g(x)=\prod _{i=1}^{n}(x-a_{i})}$ ${\displaystyle K}$ ${\displaystyle f(x)}$

To avoid such problems, one considers polynomials not only as polynomial functions with a certain domain, but tries to realize the two aspects at the same time. On the one hand, the polynomials are characterized in a reversibly unambiguous way by their coefficients, see the article on the polynomial ring. On the other hand, the functional character of the polynomials should also be preserved, in such a way that one can use elements from the bodies or rings that expand the coefficient range in polynomials . This is achieved through the insertion homomorphism, whereby a real polynomial function is created according to the pattern of the abstract polynomial. ${\displaystyle x}$

literature

• Albrecht Beutelspacher : Linear Algebra. An introduction to the science of vectors, maps, and matrices. With loving explanations, illuminating examples and worthwhile exercises, not without funny sayings, a humorous tone and slight irony, presented for the benefit of the first-semester students. 6th revised and supplemented edition. Vieweg, Braunschweig et al. 2003, ISBN 3-528-56508-X ( mathematics for first-year students ).
• Siegfried Bosch : Algebra. 7th edition. Springer-Verlag, 2009, ISBN 3-540-40388-4 , doi: 10.1007 / 978-3-540-92812-6 .
• Rolf Busam, Thomas Epp: Examination trainer for linear algebra. Spectrum, Heidelberg 2009, ISBN 978-3-8274-1976-7 .