Is there a name for these sequences of subsets of a commutative ring resembling the definition of a graded algebra? (I am experimenting with writing arrows backwards.) Let $R$ denote a commutative ring. Is there a term for those sequences $A : \mathcal{P}(R) \leftarrow \mathbb{N}$ satisfying the following requirements resembling the definition of a graded ring? 1. $A_i A_j \subseteq A_{i+j}$ and $1 \in A_0$ 2. $A_i + A_i \subseteq A_i$ and $0 \in A_i$ 3. $RA_i \subseteq A_i$ _Example._ Let $R = \mathbb{Z},$ $A_0 = \mathbb{Z}$, and $A_i = 2\mathbb{Z}$ for all $i>0$. **Motivation.** If a sequence $A$ into $\mathcal{P}(R)$ satisfies these conditions, then the following subset of the polynomial algebra $R[x]$ is in fact an $R$-subalgebra: $$\bigcup_{n \in \mathbb{N}}\sum_{i < n}A_i x^i$$

2 and 3 ensures that the $A_i$'s are (left) ideals of $R$, and 1 that $A_0 = R$. Such a sequence is called a _filtration_ of the unital ring $R$ by (left) ideals. More generally, you have the notion of _filtration_ of an (left) $R$-module by a sequence of $R$-submodules. Endowed with such a filtration, your object is called a _filtered ring_ or _filtered module_. If this filtration has some monotonicity property (like being increasing or decreasing) you can define a graded object (ring or module) associated to the filtered object.