What is ℤ2[x]/I?

Question

Sorry for asking such a "dumb" question. I'm not learning ring theory systematically. I just see such a symbol but don't understand what it means. I checked ring division. It seems that is another thing. In my case, is ℤ2[x]/I equivalent to exclude, which is all elements in ℤ2[x] - all elements in I? What do we call ℤ2[x]/I?

Answer 1

You're correct that it's not related to set difference. $\mathbb{Z}_2[x]/I$ is the quotient ring where two polynomials are equivalent if their difference is in $I$. We can write every polynomial as $p(x)+I$ for some $p(x)$ in $\mathbb{Z}_2[x]$. Then we can add and multiply using the rules:

$$(p(x)+I) + (q(x)+I) = (p(x)+q(x))+I$$ $$(p(x)+I)(q(x)+I) = p(x)q(x)+I.$$

We often drop the $I$ or use other notation once it is understood we are in the quotient ring. Note that this construction is only well defined if $I$ happens to be an ideal. Otherwise our operations might give us different outputs depending on which representative in the quotient ring we choose. There are many, representatives as $p(x)+i(x) + I = p(x) + I$ for any $i(x)$ in $I$.