Let $R$ be a commutative noetherian ring and $I=E(R/p)$ the injective hull of the module $R/p$ for a prime ideal $p$.
Question: Is there a (more) explicit description of the $R$-modules $Ext_R^i(I,R)$ for $i geq 0$ (at least in special cases such as $R$ being Gorenstein)? When are those modules finitely generated?
Let $A$ be a commutative algebra over a field $k$ which is finite dimensional as a vector space over $k$. Let $M$ be a faithful $A$-module. Does it follow that $dim_k(M)geq dim_k(A)$?
Is it true that for simple $ C ^ * $-Algebras, which means that they have no non-trivial bilateral ideals, it means that they are necessarily not commutative? And why?