\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \newcommand{\E} {\mathrm{E}} \)
deepdream of a sidewalk
Show Answer
\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \require{mathtools} \)
Math and science::Algebra::Aluffi

Prime and maximal ideals

Prime and maximal ideals

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

Prime
\( I \) is said to be a prime ideal iff \( R / I \) is [what?].
Maximal
\( I \) is said to be a maximal ideal iff \( R / I \) is [what?].

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 [ \[ \forall a, b \in R, \;\; ab \in I \implies \text{ what? } \quad\quad\quad\quad \]]
Maximal
\( I \) is a maximal ideal iff [\[ \text{for all ideals } J \subseteq R, \;\;I \subseteq J \implies \text{ what? } \quad\quad\quad\quad \]]

Can you construct the proof of this equivalence?