Is it true that any algebra is Morita equivalent to oposite?

Question

Let $A$ be an algebra with unit. Denote by $A^L$ the algebra $A$ considered as left module over itself. Then $A^{op} = End_A(A^L)$.

So according to ncatlab.org/nlab/show/Morita+equivalence $A^{op}$ is morita equivalent to $A$ (namely "The algebra $S$ is isomorphic to the endomorphism algebra of a finitely generated projective generator in the category of left (or right) $R$ -modules.")

Is it correct? If not, then what is counterexample?

Answer 1

The nLab article is incorrect, it should say "right modules" specifically, not left modules.

It is not true that an algebra is always Morita equivalent to its opposite. For every field $k$ there is a group $\text{Br}(k)$, the Brauer group of $k$, whose elements can be described as Morita equivalence classes of central simple algebras over $k$, and where the group operation is tensor product. Taking the opposite algebra gives inverses in this group. So any element of a Brauer group of order greater than $2$ gives a counterexample, and for example the Brauer group $\text{Br}(\mathbb{Q})$ is known to have elements of every finite order. I believe examples are given by cyclic algebras.

Answer 2

Is it true that any algebra is Morita equivalent to opposite?

Another way to see that this could not be so is by considering any Morita invariant property for a ring that has that property only on one side. For example, there exists a left-not-right Noetherian ring, and it's known that "left Noetherian" is a Morita invariant property (it is shared by all rings Morita equivalent to that ring.)

When $R$ is left-not-right Noetherian, its opposite ring is right-not-left Noetherian. If $R$ were Morita equivalent to its opposite, we would have a contradiction.

The source you're reading appears to say:

The algebra $R$ is isomorphic to the endomorphism algebra of a finitely generated projective generator in the category of left (or right) $R$-modules;

The algebra $S$ is isomorphic to the endomorphism algebra of a finitely generated projective generator in the category of left (or right) $S$-modules.

I think a corrected version should be:

The ring $S$ is isomorphic to the endomorphism ring of a finitely generated projective generator in the category of right $R$-modules;

The ring $R^{op}$ is isomorphic to the endomorphism ring of a finitely generated projective generator in the category of left $S$-modules.

The first one is for sure correct, and is the one usually seen in print. I don't think I've seen the second one in print but I believe it can be derived from the first. In Morita theory, you work with a special bimodule $_SP_R$ and it turns out that $P_R$ is a projective generator iff $_SP$ is a projective generator, and from there we get the first version $S\cong End(P_R)$.

Now, reversing the sides of a module action yields a module action over the opposite ring, so we could talk about the bimodule $_{R^{op}}P_{S^{op}}$, and you can regard $P_{S^{op}}$ as a f.g. projective generator for right $S^{op}$ modules. According to the half we already believe, it must be that $R^{op}\cong End(P_{S^{op}})=End(_SP)$. So that yields the corrected second version.

We can see these are both consistent with the results that $R\cong End(R_R)$ and $R^{op}\cong End(_RR)$, reflecting the mundane fact that $R$ is Morita equivalent to itself.