Linear topology

Remarks

Equivalently, a linear topology on $A$ is given by a family $\mathcal{F}$ of ideals such that each intersection $I \cap J$ of two members contains a member. A set $U$ is then open when for each $x \in U$ there is $I \in \mathcal{F}$ with $x + I \subseteq U$.

Example: for an ideal $I$ of $A$, the I-adic topology has the powers $I^n$ as a fundamental system of neighbourhoods of $0$. Each $I^n$ is an open ideal, so clause 2 holds; addition and multiplication are continuous because $(x + I^n) + (y + I^n) = x + y + I^n$ and $(x + I^n)(y + I^n) \subseteq xy + I^n$. The discrete topology is linear, with the zero ideal open.

Non-example: the usual topology on $\R$ makes $\R$ a topological ring, so clause 1 holds. But the only ideals of the field $\R$ are $0$ and $\R$. The set $\set{0}$ is not open, and the interval $(-1, 1)$ does not contain $\R$; so clause 2 fails.