Show that if b ∈ R, R is a Eucledian domain and a is a non-zero non-unit, then bR/aR = {r + Ra : r ∈ bR} is an ideal in R/aR.

Suppose that R is a Euclidean domain and a ∈ R is a non-zero non-unit.

For an element b ∈ R consider the subset bR/aR = {r + Ra : r ∈ bR} of R/aR.

Let r + Ra, s + Ra ∈ bR/aR so that r, s ∈ bR, then r + s ∈ bR which gives (r + Ra) + (s + Ra) = (r + s) + Ra ∈ bR/aR.

Let r ∈ Ra ∈ R/aR and s + Ra ∈ bR/aR so that s ∈ bR, then rs ∈ bR implies that (r + Ra) · (s + Ra) = (rs) + Ra ∈ bR/aR.

Hence bR/aR is an ideal in R/aR

Just in case you weren’t sure that it was.

It’s almost 2 am, what do you expect?

Oh, and I goofed while writing a game theory midterm last week, and only got 92/100.

I forgot to do part II of question 1, inspite of the note on the test paper saying “Don’t forget to do part (ii) below”.


Oh, and should I do a blurb on the math behind error correction, specifically, as it’s implemented on CDs?