The Existence of Rigid Systems of Maximal Size

Abstract

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 G_i (i∈p) of R-modules of power λ such that the endomorphism algebra Ena G_i coincides with a prescribed R-algebra A modulo some “inessential” endomorphisms which form an ideal Ines G_i. Furthermore Hom(G_i,G_j) consists of inessential homomorphisms only for any i &#x2260 j ∈ p. Naturally, we want ρ to be as large as possible which is ρ = 2^λ . In all “classical cases” we derived ρ = 2^λ , but it would be much nicer to obtain ρ = 2_λ without any restrictions as assumed in [CG], Theorem 5.2(b). The following theorem will settle this problem which will be our main result.