[Mathematical Linguistics] Identity Elements, Inverse Elements, and Zero Elements

We talk about identity elements, inverse elements, and zero elements in algebras for a given operation, and then relate it to the operations and sets we've seen so far in the course.

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[Intrebute]

Can we get some examples of (left | right) (identity | inverse | zero) for operations that aren't commutative?

Also, less of a question, but something that I wanted to think about for a bit.

e is identity,
0 is zero,
* is the operation
a, b, c are random elements in this:

e * a = a
a * e = a

fine here

0 * a = 0
a * 0 = 0

fine so far right?
inv(a) for inverse of element a

inv(a) * a = e
a * inv(a) = e

And now for what I wanna see filled up

f(a) * a = 0
a * f(a) = 0

what would f If not, could this be the defining equation of something interesting?