What comes next?

What comes next in the following sequences?

1. ..., 4, 14, 23, 34, 42, 50, __, 72, 81, ...
2. ..., a, f, p, n, __, m, ...
3. ..., 1, 1, 1, 1, 1, __, ....

Dumb Stuff I Didn't Know, Volume I of Many

I was reading something the other day (context largely unimportant) and found myself completely unable to understand an argument that was being made. On further inspection, I was able to ascertain that the author had felt that the following was so obvious as not to merit any mention: that diagonalizable matrices commute if and only if they are simultaneously diagonalizable, i.e. if there is some basis which is an orthogonal basis of eigenvectors for both matrices.

Trying to prove this fact once you know what you're looking for isn't too hard -- playing around with formulas will show you quickly that if two linear maps commute then they must fix each others' eigenspaces; then just diagonalize the restriction of S to each of T's eigenspaces and stick the resulting mini-bases together.

We can do a little better here: Suppose that we have some commutative collection C of endomorphisms of a finite-dimensional complex vector space V; then they all share some eigenvector.

Proof: We know that C fixes V. Let W be a subspace of V of minimal positive order which is fixed by C. We claim that each vector in W is an eigenvector of each element of C. Suppose not; then there is some counterexample Mw≠λw. M fixes W though, so the restriction of M to W is a complex linear map in its own right; in particular, it has to have some eigenvector in W, say Mx=μx. Let's have W_μ denote the μ-eigenspace of M restricted to W. Then the previous two sentences in effect say that that 0 <>μ <>μ is fixed by C -- take any w in W_μ and any N in C and notice that M(Nw) = (MN)w = (NM)w = N(μw) = μ(Nw), i.e. Nw is a μ-eigenvector of M, and since N, being after all from C, fixes W, it's an element of W_μ. But wait! Before we said that W was as small as you could get and still be fixed by C, but we've just shown that W_μis both fixed by C and smaller than W. This can't be, so our assumption must have been wrong -- W must consist exclusively of eigenvectors for every element of C.

Quick question: how many different eigenvalues can be associated with the subspace W in this proof, for any given transformation? For all of them?

Less quick question: What happens when we pass to infinitely many dimensions?

Rings

Every time I read something that pertains to any kind of ring, there's always a necessary section at the beginning stipulating that such-and-such a ring has an identity, that the mappings under consideration must preserve that identity, and so forth. Can someone fill me in as to whether there's some area of study in which people use rings without 1 so extensively that it makes up for the extreme annoyance of dealing with such requirements everywhere else? As far as I can tell, rings without 1 occur basically as often as these things called "monoids" or "groupoids" -- more than never, but certainly not often enough to merit such a basic name, or take up space in textbooks and brains. Can someone set me straight here?