So I am having a lot of trouble proving this. It was for an assignment due tuesday, but the prof said I can have a bit of extra time with this question.
Please do not give me the answer because I am sure my prof is on this website :P
okay so the question is as follows:
Let $R$ be a ring with identity. Show that the map $f: \mathbb{Z} \rightarrow R$ given by $f(k) = k1_R$ is a homomorphism.
So obviously I went about proving the axioms thinking the $k$ and $1_R$ are integer products. BUT as my prof pointed out, there not. Here is the explanation on his course webpage.
A couple of remarks about Assignment 8 problems: In exercise 25, when H'ford writes something like $k1_R$ , where k is an integer, he of course is not speaking of a PRODUCT of elements of the ring R. Given an element a of R, the inductive definition of ka, k in ZZ, in connection with the definition of the characteristic of a ring, about which you folks had an assignment problem, was this:
0a := 0_R,
and, for all k in |N, (k+1)a := ka + a.
Also, if k is a negative integer, then ka := -((-k)a).
I believe we defined finite sums inductively a long time ago. One defines 0a as above, and we could then use our earlier inductive definition and just say that, for each k in ZZ^+, $ka=\sum_{j=1}^k a$
And then define ka for negative integers k as above.
so how do I use his definition of what $k$ is and how do I apply it to my proof?