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Jul
31
comment Some doubts about right ideals of a ring
A fortiori, the axioms of ring multiplication allow you to conclude $R$ is a right $R$ module with the ring multiplication as the action. That right ideals are the same thing as right submodules follows directly from the definitions. And finally yes, the quotient being a ring necessitates $T$ being a two-sided ideal. Is there any question left?
Jul
30
revised Difference between torsion and zero divisor
added 88 characters in body
Jul
30
comment Difference between torsion and zero divisor
I get the feeling you didn't read the definitions very carefully: how could they be considered the same? They're of a similar flavor, of course... but the definitions aren't identical...
Jul
30
answered Difference between torsion and zero divisor
Jul
30
comment Proof about the difference between right and left ideals in a ring
@JavierArias I think someone has already pointed out to you in your proof that if the $R$ in the proof is commutative, it will be proving a commutative ring has a left ideal that isn't a right ideal, which is absurd. The strategy of beginning with "Let $M$ be a module over a ring $R$..." and then proving not all left ideals are right ideals is doomed to failure for that reason.
Jul
30
comment Proof about the difference between right and left ideals in a ring
@JavierArias Of course it could be trying to explain heuristically why it is not the case, but that is not really a formal proof. Similarly if you had the statement "aliens don't exist" and you wanted to show that this statement is wrong, then you can wave your hands all day with axioms showing that they could possibly exist, but that isn't convincing. What would really be convincing is to produce an alien as a counterexample to the claim.
Jul
30
comment Rotating one coordinate system about another
This is important because your question makes sense in the context of linear transformations of a vector space, but that does not appear to be what you are asking about.
Jul
30
comment Rotating one coordinate system about another
Then I guess you are working with $3\times 3$ affine transformations to accommodate translations?
Jul
30
comment Rotating one coordinate system about another
This raises several questions. The origin of $B$ isn't the same as the origin of $A$? Are these curvilinear coordinates? Apparently you are assuming that we are working in the plane?
Jul
30
comment Proof about the difference between right and left ideals in a ring
@JavierArias I think Jyrki is right. If a proposition is true (has no counterexamples) then you can try to prove it. If a proposition is false, then it has a counterexample, an no proof exists for the proposition. Here, you are just working with the proposition "every right ideal is a left ideal" Maybe there is something linguistic going on here, but I think you should spend some time seeing it from the point of view of this solution no matter what.
Jul
30
comment Is the ring of holomorphic functions on $S^1$ Noetherian?
Theorem 1 is pretty interesting, but if the author already knows it is bezout, therefore a PID and factorial, that's a little more accessible. Can't have too much variety, I guess
Jul
29
revised Rubik's Cube's Group
edited body
Jul
29
revised Rubik's Cube's Group
added 91 characters in body
Jul
29
comment Is there a name for this simple structure?
I know that pairs of spaces are used in algebraic topology, and I have discovered for myself that the category of Clifford algebras (or exterior algebras if you prefer) is governed by this pairing picture. I suppose you can also consider the category of field extensions an example (where $\alpha$ is an isomorphism of fields). What other examples did you find? I'm interested...
Jul
28
comment Quaternions and rotation
For the second thing, it changes how magnetic north moves: if you translate perpendicularly to the north-south on flat earth while oriented pointing west, the compass and orientation never change with respect to each other, but on round earth they do: the compass swings to follow magnetic north while the orientation stays pointing the same direction in 3 space
Jul
28
comment Quaternions and rotation
For one thing, when you do this step "After that this device is placed in some other point in space" it matters for your orientation after translation. If you were on a "flat earth" and oriented pointing down, then translating would never change that you are pointing down. But translating near a round earth would change your down into up after you get to the opposite side of the globe.
Jul
28
comment Quaternions and rotation
This is a dumb question, but in your model, are you assuming a realistic spherical earth for your magnetic field? Or are you approximating with flat map-like earth?
Jul
28
revised $R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$
added 4 characters in body
Jul
28
comment Comments about “Topics in Algebra” by I.N. Herstein and “Abstract Algebra” by Dummit/Foote?
@mathwanderer Posts soliciting opinions are generally not acceptable. There are exceptions probably but it is a useful rule.
Jul
28
answered $R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$