Introduction
Let A be a ring and M a (left) A-module, and let Y be the endomorphism ring
EndA(M). Then we have the natural covariant functor
HOI21A(M,
): Mod A
Modr
o p
and the natural contravariant functor HoniA( , M): Mod A » ModT.
Here Mod A denotes the category of (left) A-modules. Considering the pair (A, M)
consisting of a ring together with a module over the ring, we obtain in the first
case, by applying HomA(M, ) to A and to M, a new pair (r op , HoniA(M, A)). We
write a(A, M) = (r op ,
HOIIIA(M,
A)), and say that there is a (covariant) Wedder-
burn correspondence between the pairs if a(rop,HoniA(M, A)) is equal (isomor-
phic) to (A,M). By this we mean that the natural ring homomorphism A
Endr°p(HoniA(M, A)) is an isomorphism and via this isomorphism the natural
group homomorphism M HomroP(HomA(M,
A),HOIXLA(M,
M)) is an isomor-
phism of A-modules. It is an interesting problem to identify pairs (A, M) and
(£, N) which correspond to each other under such a (covariant) Wedderburn cor-
respondence. In this case it is of interest to identify subcategories of Mod A and of
Modr o p such that the functor HomA(M, ) induces an equivalence between them.
Note that there is always induced an equivalence HomA(M, ): addM addr o p
and HomA(M, ): add A addHoniA(M, A), where for a module X the objects of
addX are the summands of finite direct sums of copies of X.
A well known example of a (covariant) Wedderburn correspondence is when
(A, M) is a pair with M a finitely generated projective generator, which gives
rise to a Wedderburn correspondence between (A, M) and (r, HomA(M, A)),
where HomA(M, A) is a finitely generated projective generator. The functor
HomA(M, ): Mod A » Modr o p is an equivalence of categories, known as Morita
equivalence. If M is assumed to be a finitely generated generator (which is not
necessarily projective), the corresponding T-module HomA(M, A) will still be pro-
jective, but not necessarily a generator. But there is also in this case a Wedderburn
correspondence between the pairs (A, M) and (r,HomA(M, A)). Exactly which
projective modules occur here was identified in [1] in the case of artin rings, and
the terminology of Wedderburn correspondence was used in this case. In particu-
lar, by specializing further, the well known correspondence between artin rings of
finite representation type and artin rings with global dimension at most two and
dominant dimension at least two is obtained this way. Recall that A has dominant
at least two if in a minimal injective resolution 0 A » 1$ l\ I2 •, the
modules Jo
a n
d I\ are projective. In this case HomA(M, ) induces an equivalence
between Mod A and the category of projective
rop-modules.
On the other hand, when starting with a pair (A, M) and applying the con-
travariant functor HomA( , M): Mod A ModT to A and to M, we obtain the
new pair (r, M), and we write /3(A, M) = (I\ M). Having /3(r, M) equal to (A, M)
IX
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