Question

prove the left distributive law for R[x] where R is a ring and x is an indeterminant.

Answer 1

\[(\sum_{i=0}^{\infty}a _{i}x ^{i})[(\sum_{j=0}^{\infty}b _{j}x ^{j})+(\sum_{k=0}^{\infty}c _{k}x ^{k})]\]
I am applying a(b+c)=ab+ac to \[a _{i}, b _{j}, c _{k}\] in R

Answer 2

its just repeated use of the fact that the distributive property holds in \(R\) but the notation is going to get very ugly....

Answer 3

could you possibly show me just how to combine the two summations added together in the parentheses?

Answer 4

when we add polynomials we add the terms matching the indeterminant, so

\(\sum_{i=0}^\infty a_ix^i+\sum_{i=0}^\infty b_ix^i=\sum_{i=0}^\infty (a_i+b_i)x^i\)

Answer 5

no need for the i and j

Answer 6

I assume you are putting infinity because at some point we only 0's? because a polynomial with infinite degree does not make sense, we just say they are infinite sums but we truncate and call the "last" non 0 coefficient term, the degree term.... so an infinite sum still has finite degree....hope that makes sense

Answer 7

i.e. \(\sum_{i=0}^\infty a_ix^i=a_0+a_1x+a_2x^2+.....+a_nx^n+a_{n+1}x^{n=2}....\) where all \(a_i=0\) for all \(i>n\)

Answer 8

that should be n+2 not n=2.