Four submodules suffice for realizing algebras over commutative rings (Q752813)

Let R be a commutative ring with identity and for \(n\geq 0\), \(R_ n mod\) denote the category of R-modules M with n distinguished submodules. A typical object is denoted by \(M=(M,M^ k;k<n)\). The main result of this paper is the following: Let \(\lambda\) be an infinite cardinal and let A be an R-algebra generated by \(\leq \lambda\) elements. Let M be equal to \(\oplus_{\lambda}A\). Then there exists an \(R_ 4\) module \(M=(M,M^ 0,M^ 1,M^ 2,M^ 3)\), with \(A=End M\), where End M consists of all R linear endomorphisms of M which leave \(M^ k\) stable, \(0\leq k\leq 3\). As a Corollary it follows that if A is a K-algebra, of dimension \(\leq \lambda\), an infinite cardinal, where K is a field then there exists a \(K_ 4\)-module M with M of dimension \(\lambda\) with \(A\cong End M\).