The Stacks project

Exercise 111.64.1 (Definitions). Provide brief definitions of the italicized concepts.

1. a multiplicative subset of a ring $A$,

2. an Artinian ring $A$,

3. the spectrum of a ring $A$ as a topological space,

4. a flat ring map $A \to B$,

5. the height of a prime ideal $\mathfrak p$ in $A$, and

6. the functors $\text{Tor}^ A_ i(-, -)$ over a ring $A$.