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Let $R$ be a commutative ring. An internally graded $R$-algebra is an $R$-algebra $A$ for which there is a family of $R$-submodules $(A_n)_{n\in\mathbb Z}$ such that

(i) $\displaystyle A=\bigoplus_{n\in\mathbb Z} A_n$ (equality of $R$-modules);

(ii) $A_i\cdot A_j\subseteq A_{i+j}$.

A graded ideal of $A$ is an ideal $I\subseteq A$ such that the equality of $R$-modules holds:

$$I=\bigoplus_{n\in\mathbb Z} (A_n\cap I).$$

Whenever $I$ is a graded ideal we can associate a graded algebra:

$$\displaystyle \frac{A}{I}:=\bigoplus_{n\in\mathbb Z}\frac{A_n+I}{I}.$$ An externally graded $R$-algebra is a family $(A_n)_{n\in \mathbb Z}$ of $R$-modules together with bilinear and associative maps $A_i\times A_j\longrightarrow A_{i+j}$.

How to define a graded ideal and the quotient in the case of externally graded $R$-algebras?

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