Wikipedia:Reference desk/Archives/Mathematics/2022 November 22

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November 22

non Noetherian module

Can the generators of a non-Noetherian module, and everything the generators generate, be the scalars for the module? Thanks Rich (talk) 03:05, 22 November 2022 (UTC) Would that just be a ring as a module over itself?Rich (talk) 03:13, 22 November 2022 (UTC)

Isn't "everything the generators generate" the same as "all elements of the module"? Can elements of a module be its scalars? In general, the elements forming the ring of scalars may be of another type than the elements of the module. For example, if $S$ is a ring, take $M=S^{\mathbf {N} },$ the set of infinite sequences of elements of $S,$ where the group operation is componentwise addition (with the addition operator of $S$). This module has a non-finite basis of infinite unit vectors $(1,0,0,...),(0,1,0,...),(0,0,1,...),...\,.$ Let $G$ be the abelian group obtained by forgetting the module structure. To be able to use this as the scalars of a module we need to view this as a ring, so we need to add multiplication to the structure. One (possibly not unique) way is componentwise multiplication, borrowing the multiplication of $S.$ Then, somewhat obviously, using this ring to turn $G$ into an honest module gives us a module with the same elements as before, but it has a different structure.  --Lambiam 08:58, 22 November 2022 (UTC)

ok, thank you Rich (talk) 00:39, 23 November 2022 (UTC)

Rotation of a spherical triangle...

Sphere centered at the origin of radius 1. Spherical triangle with corners at (1,0,0), (0,1,0) and (0,0,1). If it is rotated 30 degrees around its center on the sphere, what are the coordinates of that circle? Similar question, if the triangle is expressed as North Pole, (0 lat, 0 long) and (0 lat, 90 long), what are the coordinates in lat long of the rotated triangle?Naraht (talk) 20:01, 22 November 2022 (UTC)

It is not clear in what plane this rotation is performed? Ruslik_Zero 20:21, 22 November 2022 (UTC)

The coordinates of which circle? Did you mean the coordinates of the corners of the rotated triangle? I assume the axis of rotation is the line connecting the origin with the point $(u,u,u),$ where $u={\tfrac {1}{3}}{\sqrt {3}}.$ A formula for the rotation matrix is given at Rotation matrix § Rotation matrix from axis and angle. In this specific case, the new coordinates are given by the three column vectors. I leave the rest as an exercise.  --Lambiam 20:55, 22 November 2022 (UTC)