Trying to understand notes, any feedback is appreciated

[PsychonautQQ]

Let 0-->M-->V-->W-->0 be an exact sequence of algebra's.
We can then see that W = V/M. (is this true? Wouldn't W = V/Im(M)?)

Then i wrote 'codimension 1 in a nilpotent algebra,' no idea why i wrote it.
Anyone, if V is a nilpotent algebra, then V^2 < V.

Let M be a comdension 1 subspace of V containing V^2, then M is an ideal. (Why is this?)

The algebra structure of W is trivial. (Why is this?)

If anyone can shed some light on any of this it would be much appreciated.

 

[mathwonk]

"
Let 0-->M-->V-->W-->0 be an exact sequence of algebra's.
We can then see that W = V/M. (is this true? Wouldn't W = V/Im(M)?)

yes but one oftenw rites V/M since in this case the map m-->V is injective because of the 0 on the left.

Then i wrote 'codimension 1 in a nilpotent algebra,' no idea why i wrote it.
Anyone, if V is a nilpotent algebra, then V^2 < V.

Let M be a comdension 1 subspace of V containing V^2, then M is an ideal. (Why is this?)

well what is the defginition of an ideal? M is an ideal if V.M is contained in M, but you just said in fact all of V.V is properly contained in M.

The algebra structure of W is trivial. (Why is this?)

Well again, you just said the product V.V is contained in M which is zero in W, so the products are all zero in W.

These are almost self explaining questions, and you probably could have figured them out yourself if you just asked yourself for the definitions of the words you were using.

If anyone can shed some light on any of this it would be much appreciated."