consider the ag theory of .... ???an epimorphism from the standard line to itself ... ??? ???something about k-module equipped with monic linear operator ... ???

how about, for example, epimorphism from standard line to direct sum of two copies of it?? ???k-module equipped with pair of linear operators which are "jointly monic" or something??? ???given x there's no more than a single pair y,z st f(y) + g(z) = x ?? ??might as well take x = 0 ??... f(y) + g(z) is never 0 unless both y and z are???

??certainly seems like a weird theory; what "use" could it possibly have???

-xg f + xf g = 0 .... ????must set (-xg,xf) to zero then??? ??but then must also set xf to zero, and thus also (x,0), and thus also x ... ???or something???

??if this is correct, then maybe it's somewhat promising that such a weird theory is syntactically inconsistent ... ???or something ??...

??but wait, what about taking natural numbers as basis, and having f map each basis element n to 2n, and g map it to 2n+1 ??? ??or something??? ... ???...

??oh, but that f and g don't commute ... hmm, interesting ... ???something about ... ??you can have a non-trivial parallel pair of endo-maps whose images intersect trivially, but not if they commute ... ??or something...