Examples of Morita equivalent rings Can someone give some examples of Morita equivalent rings different from the classical one? (i.e. that a ring $R$ is Morita equivalent to the ring $M_n(R)$)
 A: Essentially the question asks for progenerators ( = finitely generated projective generators) $P$ of $\mathsf{Mod}(R)$, since then $R$ and $S=\mathrm{End}(P)$ are Morita equivalent, and every Morita equivalence arises this way. Let us assume that $R$ has only trivial idempotents (one can assume this wlog when $R$ is noetherian). Then it is known (Lam, Lectures on Modules and Rings, 18.11) that every finitely generated projective module $\neq 0$ is already a progenerator. So the question is which are examples of finitely generated projective modules which are not free (and actually we want that their endomorphism ring is not just a matrix algebra, which singles out commutative rings and lot of other examples). These exist when the algebraic K-theory $K_0(R)$ is larger than $\mathbb{Z}$. The book Algebraic K-theory and applications contains the basic theory and many examples.
For example (see 1.2.7 in loc.cit), let $k$ be a field and consider $R=\mathrm{colim}_n ~M_{2^n}(k)$ with the transition maps $A \mapsto \mathrm{diag}(A,A)$ (this is related to the well-known CAR-algebra in functional analysis). Then $K_0(R) = \mathrm{colim}_n~ K_0(M_{2^n}(k)) = \mathrm{colim}_n ~K_0(k) = \mathrm{colim}_n ~\mathbb{Z}$ with the transition maps $2 : \mathbb{Z} \to \mathbb{Z}$, hence $K_0(R)=\mathbb{Z}[\frac{1}{2}]$. It is a good exercise to compute the endomorphism ring of the projective module corresponding to $\frac{1}{2}$.
A: Wikipedia asserts that $S$ is Morita equivalent to $R$ if and only if
$$S \cong e M_n(R) e $$
for some positive integer $n$ and idempotent matrix $e$ satisfying $M_n(R) = M_n(R)eM_n(R)$.
A: Consider the algebra of upper triangular $2 \times 2$ matrices$$\left( \begin{array}{cc} * & * \\ 0 & * \end{array} \right)$$
As a module over itself it is sum of two indecomposable modules. Denote them as $V_1$ and $V_2$ (one and two dimensional respectively). $P = V_1^{\oplus n} \oplus V_2 ^ {\oplus m}$ is a projective generator (here $n,m \geq 1$). For instance, n=2, m=1 provides $End(P)$ is isomorphic to
$$ \left( \begin{array}{ccc} * & * & 0 \\ * & * & 0 \\ * & * & * \end{array} \right)$$
