On a class of semicommutative modules.
Let \(R\) be a ring with 1, \(M\) a right \(R\)-module, and \(S=\text{End}_R(M)\) the ring of endomorphisms of \(M\). Then \(M\) is a left \(S\)-module and an (\(S\)-\(R\))-bimodule. Denote the annihilator of \(X\subset M\) in \(R\) by \(r_R(X)\), and the left annihilator of \(X\subset R\) in \(M\) by \(l_M(X)\). Similarly, \(r_M(X)\) and \(l_S(X)\) are defined. A module \(M\) is called \(S\)-Baer if \(l_S(N)=Se\) for any \(R\)-submodule \(N\) of \(M\) where \(e^2=e\in S\), \(S\)-quasi-Baer for any fully invariant submodule \(N\) of \(M\), and \(S\)-principally quasi-Baer (p.q.-Baer) for any principal submodule \(N\) of \(M\). A module \(M\) is called semicommutative if for \(m\in M\) and \(a\in R\), \(ma=0\) implies \(mRa=0\), and \(R\) is called semicommutative ring if \(R_R\) is semicommutative. Similarly, a semicommutative \(_SM\) and a semicommutative ring \(S\) are defined. Then the authors show some properties of \(S\)-semicommutative modules. Theorem 1. Let \(M\) be \(S\)-semicommutative. Then (1) \(S\) is semicommutative; (2) \(M\) is \(S\)-p.q.-Baer if and only if \(l_S(N)=Se\) for any finitely generated \(R\)-submodule \(N\) of \(M\), where \(e^2=e\in S\); and (3) if every fully invariant submodule of \(M\) is a direct summand of \(M\), then \(M\) is \(S\)-Baer. Let \(M[x]=\{\sum_{i=0}^sm_ix^i\mid s\geq 0,\;m_i\in M\}\). Then \(M[x]\) is a right \(R[x]\)-module with an obvious addition and the scalar multiplication by \((\sum_{i=0}^sm_ix^i)(\sum_{i=0}^ta_ix^i)=\sum_{k=0}^{s+t}(\sum_{i+j=k}m_ia_j)x^k\). Similarly, \(M[x]\) is a left \(S[x]\)-module. Theorem 2. Let \(M\) be \(S\)-semicommutative. Then \(M\) is \(S\)-p.q.-Baer (resp. \(S\)-Baer, \(S\)-q.-Baer) if and only if so is \(M[x]\) over \(S[x]\).
scientific article; zbMATH DE number 5603916