I know little ring theory, but have the following general question:
If $R$ is a $G$-graded ring ($R = \bigoplus_{k∈G}R_k$ where $R_iR_j ⊆ R_{i+j}$) and $I ⊆ R$ is an ideal ($I$ is closed under addition and $RI ⊆ I$), then how can one determine the grading of the quotient $R/I$ in general? What are the relevant properties of $I$?
I’m asking this question with the following examples in mind: The tensor algebra $\mathrm TV$ (which is naturally a ring) is $ℤ$-graded, and…
- The exterior algebra $\bigwedge(V) = \mathrm TV\big/⟨⊗⟩$ is the quotient of the tensor algebra over the ideal generated by elements of the form $⊗$ for $ ∈ V$. The exterior algebra itself is also $ℤ$-graded (and hence also $ℤ_2$-graded).
- The Clifford algebra $Cl(V, q) = \mathrm TV\big/⟨⊗ - q()⟩$ is a quotient with respect to an ideal which identifies vector squares $⊗$ with their “norm square” $q()$. The Clifford algebra is $ℤ_2$-graded (but not $ℤ$-graded).
I have a vague hope that there might be a sense in which, if $Γ(R)$ is the “maximal” monoid for which $R$ is $Γ(R)$-graded, then $Γ(R/I) ≅ Γ(R)/Γ(I)$ for some suitable definition of $Γ(I)$.
Is there a way to determine the grading of $R/I$ from the grading of $R$ and $I$ without resorting to inspecting $R/I$?
The property you are looking for is the ideal $I$ being homogeneous (or equivalently, graded). A homogeneous ideal is an ideal generated by homogeneous elements, i.e., elements belonging to some $R_k$. Equivalently, an ideal $I$ is homogeneous if it is $G$-graded, $I = \bigoplus_{k ∈ G} I ∩ R_k$ (see this post or proposition 2.1 for why these are equivalent).
The quotient $R/I$ by a homogeneous ideal has a natural grading: i.e., if $R$ and $I$ are $G$-graded, then $R/I$ is $G$-graded.
The exterior algebra is $ℤ$-graded since it is a quotient by a $ℤ$-graded ideal. The Clifford algebra is only $\mathbb{Z}_2$-graded because the defining ideal is only $\mathbb{Z}_2$-graded: with respect to this $\mathbb{Z}_2$-grading the ideal is homogeneously generated by $⊗ - q() ∈ (\mathrm TV)_0.$