Intuitive understanding of ideal $I = (x+1,x^2+1)$ and the quotient $\Bbb Z[x]/I$

Solution 1:

Hint $\!\bmod I=(\color{#c00}{x\!-\!a},f(x),g(x),\ldots)\!:\ \color{#c00}{x\equiv a}\,\Rightarrow\, f(\color{#c00}x)\equiv f(\color{#c00}a),\, g(\color{#c00}x)\equiv g(\color{#c00}a),\,\ldots$

therefore: $\,\ I = (x\!-\!a,f(a),g(a),\ldots),\ $ where we used the Polynomial Congruence Rule.

So, in OP: $\, \ I = (x\!+\!1,\, x^2\!+1) = (x\!+\!1,\,\color{#0a0}2)\ $ since $\,f(x)=x^2+1\,\Rightarrow\,\color{#0a0}{f(-1) = 2}$

So $\,\Bbb Z[x]/I = \Bbb Z[x]/(x\!+\!1,2)\cong \Bbb Z[x]/(x\!+\!1)/((2,x\!+\!1)/(x\!+\!1) \cong \Bbb Z/2\,$ via Third Isom. theorem.

Remark $ $ Above is a sort of ideal form of the basic step in the Euclidean algorithm for the gcd, viz. $$(h,f,g,\ldots) = (h,\, f\bmod h,\, g\bmod h,\ldots)$$

i.e. we can mod out all the other generators by any generator while preserving the ideal. More generally ideals are preserved under any unimodular transformation of the generators, which may be viewed as an ideal form of a "change of basis". The Euclidean algorithm generalizes in various ways, e.g. to Hermite (or Smith) normal forms, and other standard basis algorithms e.g. Grobner bases.

Such standard bases often yield a more "intuitive understanding" of the ideal, being "simpler" in various ways, e.g. they may be in triangular form and/or be a module basis, which makes it clear how to use the basis as effective normal-form rewriting rules (e.g. see here), and may also make it easier to deduce properties of the quotient ring.