Dual pairs of modules and intermediate matrix subrings
by
Juan Jacobo Simón Pinero
Universidad de Murcia
Coauthors: Angel del Río

Let R be a unital ring, and c, d be two infinite cardinals with d less than or equal to succ(c). Following the classical definition for vector spaces by D. Ornstein, a (c, d)-dual pair (F, N) over R is formed by a free R-module F of dimension c, and a submodule N of F^* = Hom_R(F, R) such that there is a basis B of F so that the elements of N are those f in F^* such that the set of elements x of B for which f(x) is nonzero has cardinal less that d. Denote by End^N(_R F) the ring of continuous endomorphisms of F with respect to the finite topology induced by N.

Let R be a unital ring and c be a cardinal number. We denote by RFMc(R) the ring of row finite matrices indexed by c, and by RCFMc(R) the ring of row and column finite matrices over R indexed by c. An intermediate subring of RFMc(R) is a subring Bc(R) of RFMc(R) which contains the ring RCFMc(R). We give a matrix description of the ring of continuous functions. Define the ring Ecd(R) as the elements A of RFMc(R) such that for any subset X of c with cardinality less than d the cardinality of the set of nonzero entries in the columns of A indexed by X is also less than d. We prove:

Theorem: Let R, S be rings, let c, d be infinite cardinals and B(R) and B(S) intermediate subrings. If B(R) is isomorphic to B(S) then the rings R and S are also equivalent rings and c=d.

Corollary: If an intermediate subring Bc(R) is isomorphic to RFMd(S) for any ring S and any cardinal d then Bc(R)=RFMc(R).

Theorem: The subrings Ecd(R) and Ec'd'(S) are ismorphic if and only if R and S are Morita equivalent, c=c' and (d=d' or d > c and d' > c).

Theorem: Let R, S be rings, (F, N) a (c, d)-pair and (F', N') a (c', d')-pair. The rings of continuous endomorphisms End^N(_R F) and End^N'(_S F') are isomorphic if and only if R and S are Morita equivalent, c=c' and d=d'.

Corollary: Let R and S be both basic semiperfect or commutative rings. Let (F, N) be a (c, d)-pair and (F', N' a (c', d')-pair. If the rings of continuous endomorphisms End^N(_R F) and End^N'(_S F') are isomorphic then there is a semilinear homeomorphism from F onto F' with respect to the finite topology.

Date received: December 23, 1998