Math and science::Algebra::Aluffi

Prime and maximal ideals

Let $R$ be a commutative ring. Let $I \neq (1)$ be an ideal of $R$ .

The above definition is equivalent to the following conditions.

Let $R$ be a commutative ring. Let $I \neq (1)$ be an ideal of $R$ .

Prime: $I$ is a prime ideal iff [ $\forall a, b \in R, a b \in I ⟹ what?$ ]
Maximal: $I$ is a maximal ideal iff [ $for all ideals J \subseteq R, I \subseteq J ⟹ what?$ ]

Can you construct the proof of this equivalence?

13.07.2023