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Nov
15
comment Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?
I have long noticed that one could define an associative (and commutative) product with $i^2=j^2 = -1$ and $k^2 = 1$, but wondered why I never heard about it. I appreciate @Micah 's comment above about getting zero divisors when trying to make a field out of a multiplicative group that contains a second element that squares to a positive real. Would this product I've specified also lead to the split quaternions, or something else? I'm guessing not since, if the split quaternions are isomorphic to the 2 by 2 matrix ring, they're also noncommutative.
Nov
13
comment What is the difference between ring homomorphism and module homomorphism?
I'm saying it wasn't a bad analogy, and examples help people who are trying to learn new concepts.
Nov
13
comment What is the difference between ring homomorphism and module homomorphism?
Clarifying your driving/flying analogy: Person $A$ and $B$ are both pilots (isomorphic under the structure: "having passed the flying exam"). Person $A$ and $C$ are both drivers (isomorphic under the structure: "having passed the driving exam"). But if person $B$ is not a driver, and person $C$ is not a pilot, these two are not isomorphic with respect to either structure.
Nov
13
comment Shorter proof of $R/I$ is a field if and only if $I$ is maximal
Your example gives a field, but I saw how $I + (x) = R$ meant that after sending $I$ to $0$, if I sent $x$ to $0$, everything collapsed, which meant $x$ was a unit in $R/I$. Further, $I$ needn't be maximal, i.e. this should work when $R/I$ is not a field. I think a more illuminating example is $I = 4\mathbb Z = (4)$. The lattice involves $(1), (2), (4)$, and $(3), (6), (12)$ "below" them. Sending $(4)$ to $(0)$, also sends $(12)$ there, meaning $(6)$ collapses to $(2)$ and $(3)$ collapses to $(1)$. $(2)$ is the only remaining maximal ideal and $3 + I$ isn't in it, so it's a unit.
Nov
12
comment What is the relationship between prime ideals and their generators?
When writing my question, I edited for simplicity, and removed everything about UFDs, but then tried to imply my question included UFDs when I said "what additional criteria are needed". Thanks for helping me understand generators better. I think over the weekend I'll type up a similar question just for UFDs and reference this one.
Nov
12
comment What is the relationship between prime ideals and their generators?
Something's not right about the 'visa-versa' part, but between Wikipedia, you, and then me, I'm not sure where the miscommunication / misunderstanding lies. In IDs (where irreducible is defined), every prime is irreducible. In UFDs, every irreducible is prime so they are equivalent (since every UFD is an ID). I originally removed this from my 'Known' section, but I've added it back.
Nov
12
comment What is the relationship between prime ideals and their generators?
In my mind "$\mathbb{Z}[x]$ and the like" can be very general. The way I understand things, $\mathbb{Z}$ is the free ring on $\{0,1\}$ (and thus the smallest free ring with unity). Then by adjoining indeterminates (but not necessarily allowing them to commute with existing elements) you can get any larger free ring, and then by imposing relations (by modding out ideals in that free ring), one can get any ring that exists.
Nov
12
revised What is the relationship between prime ideals and their generators?
added 105 characters in body
Nov
12
accepted What is the relationship between prime ideals and their generators?
Nov
12
comment What is the relationship between prime ideals and their generators?
Clearly my questions (about multiple generators) only apply outside a PID. I know that this example $\mathbb Z [\sqrt{-5}]$ is an ID, but not a UFD. I'm now wondering if this relies on the multiple ways to factor ... if this still holds in a UFD where prime and irreducible coincide.
Nov
12
comment What is the relationship between prime ideals and their generators?
Thanks, I've worked out everything except why $R/(p)$ must be finite for any $p \in I$. Why is that?
Nov
11
asked What is the relationship between prime ideals and their generators?
Nov
10
comment What's an example of an ideal in $\mathbb{Z}[\sqrt{-n}]$ that is not principal?
@goblin The most widely used convention is that integral domains are commutative, and domains are not. On the other hand, the most widely used convention for 'prime' is only defined in a commutative ring, and 'irreducible' is only defined in an integral domain.
Nov
8
revised How do I show a mapping is a homomorphism?
typo
Nov
5
comment When do two matrices have the same column space?
This is great(!) but I think you need to switch $\mathbb K^m$ and $\mathbb K^m$ in statements 3. Do you have a source for this that I can see the proof and cite?
Nov
5
comment Can a matrix have the same range and nullspace?
Some of your confusion might be that you've defined the range incorrectly. $R(A) = \{Ax \mid \forall x\}$. $b=0$ is in fact in the range of all $A$. ($A0=0$.)
Oct
1
comment Inverse of an invertible triangular matrix (either upper or lower ) is triangular of the same kind
I agree with @user23238. Simply stating that $I+N$ has an upper triangular inverse, is using what we're trying to prove.
Apr
3
comment How to find the number of roots using Rouche theorem?
Ah. Thanks. I kept trying to apply the theorem to $\frac{f(z)}{(z+1)^2}$ on either $B(0,1)$ or $B(0,1-\delta)$.
Apr
3
comment How to find the number of roots using Rouche theorem?
This answer is incorrect. As Ma Ming and The Substitute have pointed out, $F(z)$ still has a zero at $z=-1$, and so you cannot apply Rouche's Theorem on this boundary.
Mar
13
awarded  Critic