Typically, a prime is defined as follows: $p$ is prime iff $(p \mid xy \implies p \mid x$ or $p \mid y)$ and $p$ is not a unit or zero. But for ideals, we say the zero ideal is prime.
There is a strong correspondence between statements about primes and statements about prime ideals:
- "A non-unit is prime if $p \mid ab \implies p \mid a$ or $p \mid b$" vs. "A proper ideal is prime if $P \ni ab \implies P \ni a$ or $P \ni b$"
- "All non-zero primes are irreducible" vs. "all non-zero prime ideals are maximal"
- "Primes are only divisible by themselves and units" vs. "prime ideals are only contained by themselves and the whole ring (generated by a unit)"
- "Zero is divisible in every ring element vs. "the zero ideal is contained in every ideal"
- "In a UFD, non-zero elements can be uniquely factored into primes" vs. "in a Dedekind ring, non-zero ideals can be uniquely factored into prime ideals"
So, I feel that zero should be prime iff the zero ideal is a prime ideal. Why is there this discrepancy? I lean towards "zero is not a prime", but what are the consequences of rejecting the zero ideal as prime as well?