Semi-commutative modules and Armendariz modules.
Throughout \(R\) denotes an associative ring with identity and all modules are unitary left \(R\)-modules. A module \(M\) is called semi-commutative if it satisfies the following condition: whenever elements \(a\in R\) and \(m\in M\) satisfy \(am=0\), then \(acm=0\) for each element \(c\in R\). If \(M\) is a module and \(M[X]\) is the corresponding polynomial module over \(R[X]\), then \(M\) is called an Armendariz module if, whenever polynomials \(f(X)=\sum a_iX^i\in R[X]\), \(g(X)=\sum m_jX^j\in M[X]\) satisfy \(f(X)g(X)=0\), we have \(a_im_j=0\) for every \(i\) and \(j\). A module is said to be torsionless if it is a submodule of a direct product of copies of \(R\). A ring \(R\) is called left Gaussian if every cyclic module is Armendariz and is called left strongly Gaussian if every module is Armendariz. A commutative ring is called arithmetical if its lattice of ideals is distributive. The authors establish the equivalence of the following conditions: (i) \(R\) is a semi-commutative (Armendariz) ring; (ii) every torsionless module is semi-commutative (Armendariz); (iii) every submodule of a free module is semi-commutative (Armendariz); (iv) there exists a faithful module which is semi-commutative (Armendariz). They also prove that flat modules over semi-commutative (Armendariz) rings are semi-commutative (Armendariz) and show that a commutative ring is strongly Gaussian if and only if it is arithmetical.
scientific article; zbMATH DE number 2028777
scientific article