TITLE: Which rings are subrings of matrix rings?
QUESTION [20 upvotes]: In this question, all rings are commutative with a $1$, unless we explicitly say
so, and all morphisms of rings send $1$ to $1$.
Let $A$ be a Noetherian local integral domain. Let $T$ be a non-zero $A$-algebra
which, as an A-module, is finitely-generated and torsion-free.
Can one realise $T$ as a subring of the (not necessarily commutative)
ring $End_A(A^n)$ for some $n \ge 1$?

REPLY [6 votes]: Hello, I just want to add a few minor comments here, since this is a topic very close to my heart:
1) Finite MCM modules are not known to exist in dimension 3. What Hochster proved for equicharacteristic case and also in general dimension 3 (based on Ray Heitmann result) is that non-finitely generated MCM modules exist.  
2) If R is a N-graded domain over a perfect field of char p > 0 and R is locally Cohen-Macaulay on the punctured spectrum, then R admits a finite MCM. A proof can be found in:
http://www.math.utah.edu/vigre/minicourses/algebra/hochster.pdf 
3) A possible candidate for a counter-example is the local ring at the origin of the cone over some abelian surfaces. 
4) Many module-theoretic consequence of existence of finite MCM can be deduced from existence of non-finitely generated MCM and other approaches to the homological conjectures, so it may be helpful to what you want to do.
Cheers,