Primitively pure submodules and primitively divisible modules
Let \(R\) be an associative ring with a nonzero identity element. A proper ideal \(P\) of the ring \(R\) is said to be `right primitive' if \(P\) is the annihilator of a simple (right) \(R\)-module. A submodule \(N\) of the module \(M\) is called `primitively pure' (resp. `maximally pure') if \(NP=N\cap MP\) for every right primitive (resp. maximal) ideal \(P\) of the ring \(R\). A module \(M\) is said to be `primitively divisible' (resp. `maximally divisible') if \(M=MP\) for every right primitive (resp. maximal) ideal \(P\) of the ring \(R\). A ring \(R\) is called a `right max ring' if every nonzero right \(R\)-module has a maximal submodule. A ring \(R\) is said to be `right quasi-invariant' if all its maximal right ideals are ideals. A ring \(R\) is called a `PI ring' (or a `ring with polynomial identity') if \(R\) satiesfies the polynomial identity \(f(x_1,\dots,x_n)=0\), where \(f(x_1,\dots,x_n)\) is a polynomial in noncommutative indeterminates with coefficients in the ring of integers \(\mathbb{Z}\), and \(\mathbb{Z}\) coincides with the ideal generated by the coefficients of the polynomial \(f(x_1,\dots,x_n)\). The main results of the paper are Theorems 1 and 2. Theorem 1. Let \(R\) be either a PI ring or a right quasi-invariant ring. Then the following conditions are equivalent: (1) \(R\) is a regular ring; (2) all submodules of any right \(R\)-module are primitively pure; (3) all submodules of any right \(R\)-module are maximally pure; (4) all principal right ideals of the ring \(R\) are primitively pure; (5) all principal right ideals of the ring \(R\) are maximally pure. Theorem 2. For a PI ring \(R\), the following conditions are equivelent: (1) all prime ideals of the ring \(R\) are maximal ideals, and the Jacobson radical \(J(A)\) is a right or left \(t\)-nilpotent ideal; (2) there do not exist nonzero primitively divisible right \(R\)-modules; (3) there do not exist nonzero primitively divisible left \(R\)-modules; (4) there do not exist nonzero maximally divisible right \(R\)-modules; (5) there do not exist nonzero maximally divisible left \(R\)-modules; (6) \(R\) is a right max ring. (7) \(R\) is a left max ring. The paper also contains some other assertions on primitively (resp. maximally) pure submodules and primitively (resp. maximally) divisible modules. In particular some results of \textit{A. I. Kornev} [in: Universal algebra and its applications (Russian), Peremena, Volgograd 144-152 (2000); in: Abelian groups and modules (Russian), Tomsk University, Tomsk 15, 30-37 (2000)] and of \textit{V. V. Ovchinnikov} [in: Abelian groups and modules (Russian), Tomsk University, Tomsk 15, 46-54 (2000)] on maximally pure and maximally divisible modules are generalized, but at the same time ideas and techniques of the above mentioned papers are used.
scientific article; zbMATH DE number 1777804