When do maximal submodules force linearity?
All rings \(R\) considered have an identity and are not division rings, while all \(R\)-modules considered are left \(R\)-modules which are unitary, faithful and not cyclic. Maximal submodules of \(R\)-modules are assumed to be non-zero modules and if \(X\) is a subset of an \(R\)-module \(V\) and \(v\in V\), then the notation \((X:v)=\{r\in R:rv\in X\}\) is used. Further, \(\text{Soc}(V)\) denotes the socle, i.e. the sum of all simple submodules, of \(V\). Let \(R\) be a ring and \(M\) a maximal submodule of an \(R\)-module \(V\). Then \(M\) is said to force linearity on \(V\) if every map \(f\colon V\to V\) with \(f(rv)=rf(v)\) for all \(r\in R\) and all \(v\in V\) and \(f(m+n)=f(m)+f(n)\) for all \(m,n\in M\) is an \(R\)-endomorphism of \(V\). First, a rather technical but general characterization of maximal submodules that force linearity is given (Theorem 2.2). For special types of rings this characterization leads to several results such as the following ones. If \(R\) is a local ring and \(V\) is an \(R\)-module, then every maximal submodule forces linearity on \(V\) if \(\text{Soc}(V)=\{0\}\) while otherwise no maximal submodule forces linearity on \(V\) (Theorem 3.1). If \(R\) is a commutative ring, \(V\) is an \(R\)-module, \(M\) a maximal submodule of \(V\), and \(V=M+Rv\) for some \(v\in V-M\), then the following statements are equivalent: (i) \(M\) does not force linearity on \(V\), (ii) there exists \(w\in V\) such that \(\text{Ann}_R(w)=(M:v)\), and there exists \(m\in M\) such that \((Rv:m)\subseteq(M:v)\) (Theorem 3.2). If \(R\) is a commutative ring, \(P\) a prime ideal of \(R\) and \(V\) is an \(R\)-module, then \(V_P\) denotes the localization of \(V\) by the prime \(P\). Suppose \(R\) has the ascending chain condition on annihilator ideals, \(V\) is an \(R\)-module and \(M\) a maximal submodule of \(V\). Then the following statements are equivalent: (i) \(M\) forces linearity on \(V\), (ii) for every prime ideal \(P\) of \(R\), \(M_P\) forces linearity on \(V_P\), (iii) for every prime ideal \(P\) of \(R\), \(V_P\) is cyclic or \(V_P=M_P\) or \(\text{Soc}(V_P)=\{0\}\) (Theorem 3.8). In a final section, results are obtained under restrictions on the modules rather than on the rings.
ascending chain condition on annihilator ideals
scientific article; zbMATH DE number 1340436