The Existence of Rigid Systems of Maximal Size
In [CG] we have not been able to answer a question on the maximal size of (generalized) rigid systems of R-modules over commutative rings; see Remark (5.3) in [CG]. A (generalized) rigid system at the cardinal ρ will be a set Gi (i∈p) of R-modules of power λ such that the endomorphism algebra Ena Gi coincides with a prescribed R-algebra A modulo some “inessential” endomorphisms which form an ideal Ines Gi. Furthermore Hom(Gi,Gj) consists of inessential homomorphisms only for any i ≠ j ∈ p. Naturally, we want ρ to be as large as possible which is ρ = 2λ . In all “classical cases” we derived ρ = 2λ , but it would be much nicer to obtain ρ = 2λ without any restrictions as assumed in [CG], Theorem 5.2(b). The following theorem will settle this problem which will be our main result.
KeywordsAbelian Group Commutative Ring Endomorphism Ring Rigid System Endomorphism Algebra
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