Countable Dimensional Right Perfect Ring

We prove that a right self right perfect algebra which is at most countable dimensional modulo their Jacobson radical is right artinian.


Introduction
In this note we present a proof for algebra R over a field K which is at most countable dimensional modulo their Jacobson radical i.e.R/Rad(R) is at most 0 Ν This includes for example the important situation when R/ RadR is not only semisimple but also finite dimensional' Let R be a ring with identity and RadR is its Jacobson radical.If R is right perfect (i.e.R/ RadR is semisimple and RadR is left T-nilpotent) then

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R/ RadR is semisimple and hence (R/RadR) R and R (R/ RadR) are semisimple right and left R-module.Note that since R/RadR is semisimple then an R-module M is semisimple iff it is cancelled by RadR.Indeed every simple is cancelled by RadR and if RadM=0 then M has R/RadR -module structure which is semisimple ,therefore M is semisimple as the lattice of R-submodules and R/RadR sub-modules coincide in this case.
(R/RadR) R is semisimple and finitely generated, so it has a composition series i.e.
Then we can find E (M) and E (N) injective hulls of M, N contained in eR and we obtain eR= E(M) + E(N) nontrivial decomposition that is a contradiction.Note also that if R is right self injective right perfect then for each simple Right R-module, the left module Hom(S, R) is simple.First Hom (S, R) is non zero.Looking for the isomorphism types of indecomposable modules eR, these are projective, local and the cover of some simple R-module.The number of isomorphism types of such modules equals the number of isomorphism types of simple modules equal t, say.Moreover since the indecomposable eR's also injective with simple socle we see that they are isomorphic if and only if their socle isomorphic.This shows that the distinct types of isomorphism of simples occurring as socle of some eR is also t, and so each simple S must appears as a socle of some eR( i.e. it embeds in R) this shows that 0 ,this shows that ) , ( R S Hom is generated by 0 ≠ f so it is simple. In particular since each simple module embeds in R which is right self injective, it follows that R is injective cogenerator in the category of right R-module, i.e. it is a right PF (Pseudo-Frobonius) ring.
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Main Results
Let s be a set of representatives' for simple right R-modules and let t=|s| and ≅ as left R-module .This shows that all types of isomorphism of simple left R-modules are found among components of ) , ( R W Hom and the statement is proved Note: we note that the above proof further shows that there is an exact sequence of left R-modules 0 ) , ( 0 This shows that W is also semisimple as a left R-module , i.e. the right socle of R is contained in the left socle, and hence the left and the right socle or the right PF ring coincide.For a right R-module M, denote ) , ( R M Hom M = * this is a left R-module.

Proposition 2
Let R be a right injective ring and let M be a right R-module such that there is an exact sequence 0 0 With S, L simple modules and assume S= socle (M), is an exact sequence, so we get the exact sequence, 0 which is left semisimple module since it is cancelled by RadR .
Now since M has simple socle, and its socle embeds in R which is injective it follows that M embeds in R. We note that * M is generated by any ⊥ ∉ S f which will show that * M is cyclic.Indeed such f must be a monomorphism and given any other

Theorem
Let R be a right self injective right perfect algebra such that the dimension of each simple R-module is at most countable (equivalently the dimension of RadR R / is at most countable) hence R is countable, and hence right artinian infinite.We note also that if k ∑ is the k' th socle then the on k.If this is true for k , then there is an embedding infinite cardinal.So we have the following theorem: R.Since each indecomposable module (R e) has simple socle, we have Length (W) equals the number of terms in the indecomposable decomposition Re