Confusions in left and right division for Ore polynomials

[kwankyu]

sage: k.<a> = GF(5^3)
....: Frob = k.frobenius_endomorphism()
....: S = OrePolynomialRing(k, Frob, 'x')
....: S
Ore Polynomial Ring in x over Finite Field in a of size 5^3 twisted by a |--> a^5
sage: x = S.gen()
sage: f = x*a + a*x^2+a*x^4
sage: g = (a^7+a)*x^2*(a+1)+ (a^3+1)*x + 1
sage: f
a*x^4 + a*x^2 + (2*a^2 + 4*a + 4)*x
sage: g
(3*a^2 + 2*a + 4)*x^2 + (2*a + 3)*x + 1
sage: q = f // g
sage: r = f % g
sage: f == g*q+r
False
sage: f == q*g+r
True

According to the documentation (1)

The operators // and % give respectively the quotient and the remainder of the right euclidean division

But the definition of the right euclidean division is (2)

Apparently the two docs are inconsistent. What is the correct definition of the right euclidean division? @xcaruso

According to Ore himself, the right-hand division of $F$ by $G$ is $F=QG+R$. So (2) is wrong.

[xcaruso]

You're right. The documentation (2) is wrong. Thanks for catching this!

[kwankyu]

Is this correct "Right(-hand) gcd is computed by right(-hand) euclidean algorithm that does right(-hand) divisions"?

This "right" and "left" confusion seems not uncommon in papers on Ore polynomials. The wrong documentation (2) in sage makes the situation worse.

[xcaruso]

Is this correct "Right(-hand) gcd is computed by right(-hand) euclidean algorithm that does right(-hand) divisions"?

I think it's correct, yes! But all of these correspond to left lcm and left modules...

[kwankyu]

Thanks.

By left modules, you mean left ideals in the form Re for an element e of the ring R.

I will prepare a PR later.

[xcaruso]

By left modules, you mean left ideals in the form Re for an element e of the ring R.

Yes.

I will prepare a PR later.

Thanks!