4 ms·

Pretty much most sources, actually. When actually working with ideals, there's almost never an advantage to consider the whole ring an ideal. So all ideals in v

Pretty much most sources, actually. When actually working with ideals, there's almost never an advantage to consider the whole ring an ideal. So all ideals in virtually all the literature I've ever read were taken to be proper, i.e. proper subsets

WCSTombs 1y ago
I very strongly disagree. Without considering the whole ring to be an ideal, you can't even define "the ideal generated by some elements" because there may not be such an ideal, since it could be the whole ring. Likewise, you can't perform common operations on ideals like sums because they could result in the whole ring. In fact, I would say there is almost no advantage in excluding the ring itself from the set of ideals. Just to check, I have three math textbooks from my college days that include the definition of an ideal, and none of them attempt to exclude the ring itself from the definition. The obvious compromise is to introduce the concept of a proper ideal as an ideal that is a proper subset, and to use that when you need to exclude the ring itself. E.g., a maximal ideal is a proper ideal that is maximal with respect to inclusion.
- kevinventullo 1y ago
  Too late to edit, but you’re right I was misremembering. I was thinking of whether (1) should be considered a prime ideal.
Sniffnoy 1y ago
That's not my experience at all as a mathematician. "Maximal ideal" implicitly means "maximal proper ideal", yes, but generally ideals don't have to be proper unless specified so. If you don't include the whole ring as an ideal, you can't even define ideal addition, etc. I took an algebra class once from a professor who decided to define "ideal" to mean "proper ideal". After a few weeks he had to give it up because it just became too much trouble for reasons like that; he had to too often say "possibly improper ideal", i.e., this convention had the opposite effect he intended! I can't think of any other source I've seen use that convention.

WCSTombs 1y ago

I very strongly disagree. Without considering the whole ring to be an ideal, you can't even define "the ideal generated by some elements" because there may not be such an ideal, since it could be the whole ring. Likewise, you can't perform common operations on ideals like sums because they could result in the whole ring. In fact, I would say there is almost no advantage in excluding the ring itself from the set of ideals. Just to check, I have three math textbooks from my college days that include the definition of an ideal, and none of them attempt to exclude the ring itself from the definition. The obvious compromise is to introduce the concept of a proper ideal as an ideal that is a proper subset, and to use that when you need to exclude the ring itself. E.g., a maximal ideal is a proper ideal that is maximal with respect to inclusion.

kevinventullo 1y ago
Too late to edit, but you’re right I was misremembering. I was thinking of whether (1) should be considered a prime ideal.

Sniffnoy 1y ago

That's not my experience at all as a mathematician. "Maximal ideal" implicitly means "maximal proper ideal", yes, but generally ideals don't have to be proper unless specified so. If you don't include the whole ring as an ideal, you can't even define ideal addition, etc. I took an algebra class once from a professor who decided to define "ideal" to mean "proper ideal". After a few weeks he had to give it up because it just became too much trouble for reasons like that; he had to too often say "possibly improper ideal", i.e., this convention had the opposite effect he intended! I can't think of any other source I've seen use that convention.