MHB Polynomials and Polynomial Functions in I_m = Z/mZ

[Math Amateur]

I am reading Joseph J.Rotman's book, A First Course in Abstract Algebra.

I am currently focused on Section 3. Polynomials

I need help with the a statement of Rotman's concerning the polynomial functions of a finite ring such as $$ \mathbb{I}_m = \mathbb{Z}/ m \mathbb{Z} $$

The relevant section of Rotman's text reads as follows:https://www.physicsforums.com/attachments/4530In the above text we read the following:

" ... ... The reader should realize that polynomials and polynomial functions are distinct objects. For example, if $$R$$ is a finite ring (e.g. $$\mathbb{I}_m$$), then there are only finitely many functions from $$R$$ to itself; a fortiori, there are only finitely many polynomial functions. ... ... "

My question is as follows:

How, exactly (indeed, rigorously and formally) can we demonstrate that only finitely many functions from $$R$$ to itself implies that there are only finitely many polynomial functions ... ... indeed, how would we prove this statement ...Hope someone can help ... ...

Peter

 

[caffeinemachine]

I am reading Joseph J.Rotman's book, A First Course in Abstract Algebra.

I am currently focused on Section 3. Polynomials

I need help with the a statement of Rotman's concerning the polynomial functions of a finite ring such as $$ \mathbb{I}_m = \mathbb{Z}/ m \mathbb{Z} $$

The relevant section of Rotman's text reads as follows:In the above text we read the following:

" ... ... The reader should realize that polynomials and polynomial functions are distinct objects. For example, if $$R$$ is a finite ring (e.g. $$\mathbb{I}_m$$), then there are only finitely many functions from $$R$$ to itself; a fortiori, there are only finitely many polynomial functions. ... ... "

My question is as follows:

How, exactly (indeed, rigorously and formally) can we demonstrate that only finitely many functions from $$R$$ to itself implies that there are only finitely many polynomial functions ... ... indeed, how would we prove this statement ...Hope someone can help ... ...

Peter

If a set i finite then each of its subsets is also finite. Here we have the set of all the functions mapping $R$ into $R$ finite if $R$ is finite. A polynomial function is, in particular, a function.