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

a

*multiplicative subset*of a ring $A$,an

*Artinian ring*$A$,the

*spectrum of a ring*$A$ as a topological space,a

*flat ring map*$A \to B$,the

*height*of a prime ideal $\mathfrak p$ in $A$, andthe functors

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

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