4 ms·

I like the following definition: p is a prime in a ring R iff the ideal generated by p is a prime ideal of R. Equivalently, p is prime iff p | ab implies p | a

I like the following definition: p is a prime in a ring R iff the ideal generated by p is a prime ideal of R. Equivalently, p is prime iff p | ab implies p | a or p | b. The only issue with this definition is that you're forced to admit that 0 is prime in integral domains.

tacomonstrous 8y ago
It's the same issue though, since you have to specify that prime ideals cannot be the whole ring to avoid 1 being prime.
vram22 8y ago
>p is a prime in a ring R That reminds of the topic of groups, rings and fields that I had in either last year of high school or an early year of college (Math.). I thought that was a really cool topic. Not too hard (for the stuff we covered), and still some cool results/theorems in it. Also, they underlie a lot of other areas of mathematics. https://en.wikipedia.org/wiki/Group_(mathematics) https://en.wikipedia.org/wiki/Group_(mathematics) https://en.wikipedia.org/wiki/Ring_(mathematics) https://en.wikipedia.org/wiki/Ring_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics)