I am comparing theorems on normal subgroup and ideal from Fraleigh's, and come to this strange intuition. I hope my conclusion does not screw up, I hope I won't get ridiculed:
Theorem 15.18: $M$is a maximal normal subgroup of $G$ if and only if $G/M$ is simple.
To me this theorem makes sense because in $G/M$, the $M$ has been "collapsed" into either $0$ or $e$ (borrowing from Fraleigh's term.) In other words, the maximal normal subgroup $M$ has been "modded out" of $G$ such that all that is left is a simple group. Having said that, let's us now go to the second theorem:
Theorem 27.9: (Analogue of Theorem 15.18) Let $R$ be a commutative ring with unity. Then $M$ is a maximal ideal of $R$ if and only if $R/M$ is a field.
Since $R$ becomes a field only after it is "modded out" of the ideal $M$, may I thus conclude that ideal can intuitively be seen as an "anti-field," meaning that each and every element of an ideal does not have multiplicative inverse, whereas each and every element of field has multiplicative inverse?
Thank you for your time and effort.
POST SCRIPT: I found another theorem of similar flavor:
An ideal $I$ of $R$ is prime if and only if $R/I$ is an integral domain.
In similar vein, may I conclude that each element of prime ideal has zero divisor? This 4-year-old posting here strikes me as relevant to my conclusion. Thanks again.