The construction of mixed modules from a theorem of linear algebra
Let R be a commutative ring with \(0\neq 1\in R\) and \(1\in S\subseteq R\) a countable, multiplicatively closed subset of R without zero divisors. We say that an R-module M is S-torsion if \(t_ SM=M\) and M is S-torsion- free if \(t_ SM=0\), where \(t_ SM=\cup_{s\in S}M[s]\), \(M[s]=\{m\in M|\) \(ms=0\}\). We call M S-reduced, if \(0=\cap_{s\in S}Ms\). Let us denote the obvious localization of R by \(S^{-1}R\). Let (1) A be an S-torsion-free R-algebra, where S-torsion-free refers to the R-module structure of A. We choose a fixed set of n multiplicatively closed subsets \(S_ i\) of S \((i<n)\) \(with\) (2)As\(+At=A\) for all \(s\in S_ i\), \(t\in S_ j\), \(i\neq j<n.\) Further, suppose that (3) A is \(S_ i\)-reduced for all \(i<n.\) Finally (4) Each \(s\in S\) is a product of elements in \(\cup_{i<n}S_ i.\) If k is a (finite or infinite) cardinal, then the authors construct in their main theorems (2.2, 2.3, \(2.3^*)\) families \(F=F(k)\) of mixed R- modules G with additional properties: (5) (a) There exists a direct sum \(T=\oplus_{i<n}T_ i\) of n direct sums \(T_ i\) of \(S_ i\)-bounded (cyclic A-)modules such that each \(G\in F\) has \(t_ sG=T.\) (b) G is S-reduced and \(G/T=\oplus_ kS^{-1}A\) is a free \(S^{-1}A\)- module of rank k. (c) End G\(=A\oplus Hom_ R(G,T).\) (d) If G,H\(\in F\), then \(Hom(G,T)=Hom(H,T)\) and End \(G\cong End H\) as R- algebras. (e) If \(G\neq H\in F\), then \(Hom(G,H)=Hom(G,T).\) Theorem 2.2. Let k be an infinite cardinal. If A is an R-algebra as above, and R has \(n\geq 5\) sets \(S_ i\) (respectively, \(A=R\) and \(n\geq 4)\), then there exists a family \(F=F(k)\) of size \(2^ k\) satisfying (5). A set \(X\subseteq A\) is a quasi-generator of the algebra A if \(Hom_ R(A/A(X),A)=0\) for the subalgebra A(X) of A generated by X. Let q(A) denote the minimal cardinal \(| X|\) of all possible quasi- generators X of A. Theorem 2.3. Suppose k is a finite cardinal and A an R-algebra as above. a) Let \(k=2\), \(A=R\), and \(n\geq 3\). b) Let \(k>2\), \(A=R\), and \(n\geq 4\). c) Let \(k\geq (q(A)+1)\cdot rk(A)\) and \(n\geq 5\). Then in each case there exists \(F=\{G\}\) satisfying (5). Theorem \(2.3.^*\) Suppose k is a finite cardinal and A an R-algebra as above. a) Let \(k=2\), \(A=R\), and \(n\geq 4\). b) Let \(k>2\), \(A=R\), and \(n\geq 5\). c) Let \(k\geq (q(A)+1)\cdot rk(A)\) and \(n\geq 6\). Then in each case there exists a family F satisfying (5) and \(| F| =| S^ 1|\), where \(S^ 1\) is the set of nonzero divisors in R. Let R be a Dedekind domain. Define the torsion submodule tG of the R- module G to be \[ tG=\{g\in G| \quad \exists 0\neq r\in R,\quad gr=0\}=\oplus_{P\in mspec(R)}t_ PG. \] From the main theorems the authors obtain corollaries (0,1, 0,2) giving an answer in uninvestigated cases to the following two problems concerning R-modules G, \(G_ i\) (i\(\in \{0,1\}):\) (*) End \(G_ 0\cong End G_ 1\) implies \(G_ 0\cong G_ 1.\) (**) Let A be an R-algebra. Find G with End G\(=A\oplus Hom(G,tG).\) Taking into consideration the results obtained for (*) the authors assume (k) \(G_ 0/tG_ 0\cong G_ 1/tG_ 1\cong \oplus_{k}Q\) where Q is the quotient field of R. Corollary 0.1. Let R be a Dedekind domain with \(| mspec(R)| \geq 4\). Then there exist at least \(| R|\) non-isomorphic R-modules \(G_ i\) with the same endomorphism algebras, such that (1) \(G_ i/tG_ i=Q\oplus Q\), (2) \(tG_ i\) is a fixed direct sum \(T_ 0\oplus...\oplus T_ 3\) of \(P_ i\)-modules \(T_ i\) \((P_ i\in mspec(R))\) which are direct sums of bounded R-modules. Q\(\oplus Q\) in (1) may be replaced by any infinite cardinal k. In this case there exist \(2^ k\) non-isomorphic examples. Corollary 0.2. Let k be any infinite cardinal, R a Dedekind domain, and A a torsion-free R- algebra which is P-reduced \((\cap_{n\in \omega}AP^ n=0)\) for at least five different primes \(P\in mspec(R)\). Let \(R^ 1\) be the localization of R at the complement of the five primes. Then there exists a torsion module T which is a direct sum of bounded R-modules for the five primes and a family of \(2^ k\) R-modules \(G_ i\) such that for any \(i\neq j\in 2^ k\) (1) \(G_ i/tG_ i=\oplus_ kQ\), (2) \(tG_ i=T\), (3) End \(G_ i=(A\otimes R^ 1)\oplus Hom(G_ i,T)\cong End G_ j\), (4) \(Hom(G_ i,G_ j)=Hom(G_ i,T)\). In particular, \(G_ i\not\cong G_ j\). A similar result holds for finite \(k\geq 2\). Condition (0.2) (4) leads to counterexamples for Kaplansky's test problems for mixed modules. The methods of the proof differ completely from the known constructions of mixed modules. They are based on a theorem of linear algebra, realizing Q on a vector space as the subalgebra of all linear maps leaving four or five subspaces fixed.
scientific article; zbMATH DE number 4023569
scientific article