Pure Baer injective modules.

*(English)*Zbl 0886.16003Let \(R\) be a ring with identity. All modules are unitary left \(R\)-modules and all homomorphisms are \(R\)-homomorphisms. A left \(R\)-module \(M\) is called pure-split if every pure submodule of \(M\) is a direct summand. \(R\) is said to be left pure split if \(_RR\) is pure-split. A non-zero \(R\)-module \(M\) is called pure simple if \(\{0\}\) and \(M\) are its only submodules.

The author gives the following definitions: 1) An \(R\)-module \(M\) is called a pure Baer injective module if for each pure left ideal \(I\) of \(R\), any \(R\)-homomorphism \(f\colon I\to M\) can be extended to an \(R\)-homomorphism \(\overline f\colon R\to M\); 2) A ring \(R\) is called left pure hereditary if every pure left ideal of \(R\) is projective; 3) A ring \(R\) is called an SSBI-ring if every semisimple \(R\)-module is pure Baer injective; 4) A left \(R\)-module \(M\) is called \(\Sigma\)-pure Baer injective if every direct sum of copies of \(M\) is pure Baer injective.

Some of the main results established by the author are: 1) The following statements are equivalent: (i) \(R\) is left pure hereditary. (ii) The homomorphic image of a pure Baer injective \(R\)-module is pure Baer injective. (iii) Any finite sum of injective submodules of an \(R\)-module is pure Baer injective; 2) Every direct sum of copies of \(R\) is pure-split if and only if every flat \(R\)-module is pure-split; 3) Let \(M\) be an \(R\)-module in which every cyclic submodule is pure-split, then every non-zero submodule of \(M\) contains a pure-simple submodule; 4) A ring which is both a \(V\)-ring and an SSBI-ring satisfies the ascending chain condition on pure left ideals; 5) A ring \(R\) in which every injective \(R\)-module is \(\Sigma\)-pure Baer injective, satisfies the ascending chain condition on pure left ideals.

The author gives the following definitions: 1) An \(R\)-module \(M\) is called a pure Baer injective module if for each pure left ideal \(I\) of \(R\), any \(R\)-homomorphism \(f\colon I\to M\) can be extended to an \(R\)-homomorphism \(\overline f\colon R\to M\); 2) A ring \(R\) is called left pure hereditary if every pure left ideal of \(R\) is projective; 3) A ring \(R\) is called an SSBI-ring if every semisimple \(R\)-module is pure Baer injective; 4) A left \(R\)-module \(M\) is called \(\Sigma\)-pure Baer injective if every direct sum of copies of \(M\) is pure Baer injective.

Some of the main results established by the author are: 1) The following statements are equivalent: (i) \(R\) is left pure hereditary. (ii) The homomorphic image of a pure Baer injective \(R\)-module is pure Baer injective. (iii) Any finite sum of injective submodules of an \(R\)-module is pure Baer injective; 2) Every direct sum of copies of \(R\) is pure-split if and only if every flat \(R\)-module is pure-split; 3) Let \(M\) be an \(R\)-module in which every cyclic submodule is pure-split, then every non-zero submodule of \(M\) contains a pure-simple submodule; 4) A ring which is both a \(V\)-ring and an SSBI-ring satisfies the ascending chain condition on pure left ideals; 5) A ring \(R\) in which every injective \(R\)-module is \(\Sigma\)-pure Baer injective, satisfies the ascending chain condition on pure left ideals.

Reviewer: I.Crivei (Cluj-Napoca)

##### MSC:

16D50 | Injective modules, self-injective associative rings |

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

16P70 | Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras) |