Semisimple implies zero-dimensional

This article gives the statement and possibly, proof, of an implication relation between two commutative unital ring properties. That is, it states that every commutative unital ring satisfying the first commutative unital ring property must also satisfy the second commutative unital ring property
View all commutative unital ring property implications | View all commutative unital ring property non-implications |Get help on looking up commutative unital ring property implications/non-implications
|

Statement

Verbal statement

If a commutative unital ring is semisimple (i.e. its global dimension is zero, or equivalently, every module over it is semisimple) then it is zero-dimensional: every prime ideal in it is maximal.

Proof

Fill this in later