[Sharlin]

Why is "commutative +" a step up rather than a step to the right? I guess there should be Abelian groups and commutative rings somewhere between groups and modules.

[ivanbakel]

Probably because the diagram originated in a Vector Spaces book, and commutativity is viewed more as a valuable property than a structural constraint.

Do physicists have any use for non-commutative algebra? It already seems pretty niche in mathematics.

[QuesnayJr]

All of quantum mechanics is non-commutative algebra. The commutative relation [x, p] = i*hbar gives you the Weyl algebra, for example.

[sergius]

Rotation group maybe? And those gauge symmetries... heck, just look into quantum mechanics operators.