RD-flatness and RD-injectivity

It is proved that every commutative ring whose RD-injective modules are Σ-RD-injective is the product of a pure semi-simple ring and a finite ring. A complete characterization of commutative rings for which each artinian (respectively simple) module is RD-injective, is given. These results can be obtained by using the properties of RD-flat modules and RD-coflat modules which are respectively the RD-relativization of flat modules and fp-injective modules. It is also shown that a commutative ring is perfect if and only if each RD-flat module is RD-projective. RD-purity is an important example of relative purity. It is the first notion of purity (for submodules) that appeared in the mathematical literature. Another reason of this importance is that for some classes of rings, not necessarily commutative, RD-purity coincides with purity. In [14] G. Puninski, M. Prest and P. Rothmaler studied these rings that they called RD-rings. In particular they proved that the classe of commutative RD-rings is exactly the classe of arithmetic rings, i.e the rings with a distributive lattice of ideals. The first section of this paper is devoted to the study of RD-flat modules and RD-coflat modules which are respectively the RD-relativization of flat modules and fp-injective modules. It is interesting to note that the RD-flat modules form an example of additive accessible category by Proposition 1.1.5. Such categories were studied by Crawley-Boevey in [5], using the term of locally finitely presented additive categories. This allows us to prove that a commutative ring is perfect if and only if each RD-flat module is RD-projective (Corollary 3.3). On the other hand, it is shown (Theorem 1.4) that a ring is RD if and only if any right module is RD-flat (or RD-coflat). So, we can say that each RD-ring is absolutely RD-flat. Recall that a ring is said to be pure semi-simple if each right module is pureinjective. It is known that a ring R is pure semi-simple if and only if each right pure-injective R-module is Σ-pure-injective. In section 2 (Theorem 2.1) we proved the RD-variation of this result: any right RD-coflat R-module is RD-injective if and only if any right RD-injective R-module is Σ-RD-injective. Moreover, any commutative ring which enjoys these properties, is the product of a pure semisimple ring and a finite ring. By [10, Observation 3(4) and Theorem 6] every artinian module over a commutative ring is Σ-pure-injective. However, [4, Example 4.6] is an example of simple module over a noetherian domain that fails to be RD-injective. If R is a commutative artinian ring, it is proved (Proposition 1.2) that R is an RD-injective module if and only if R is quasi-Frobenius. In section 4 we give a characterization of commutative rings for which any simple module is RD-injective (Theorem 4.1) and a characterization of commutative rings whose artinian modules are RD-injective 2000 Mathematics Subject Classification. 16D80, 16P20, 16P70, 13E10.