Relative algebraic structures.(English)Zbl 1067.20500

Summary: The concept and some of the algebraic properties of the rejective and non-absorptive sets of a subgroup, subring, and subgroup of a module over a ring are investigated. It is shown that the set theoretic complement of a non-absorptive set in the above mentioned algebraic substructures is a normal subgroup (respectively, (left, right) ideal, submodule) of its underlying algebraic structure. The invariant property of the non-absorptive sets under the operation of inversion in the related underlying algebraic structure is proved. $$G\setminus R(H)$$, the set theoretic complement of the rejective set of a subgroup $$H$$ in a group $$G$$, is closed under the product in $$G$$ and whenever $$|G|$$ the order of the group $$G$$ is finite, $$|R(H)|=(k-s)|H|$$ where each of the $$k$$ and $$s$$ is the index of $$H$$ in $$G$$ and in $$G\setminus R(H)$$, respectively. For the case of rings and modules, the set theoretic complement of the rejective set of a substructure in the underlying ring is a subring of the underlying ring. For any subring $$S$$ of a ring $$R$$, examples and some of the properties of $$S$$-relative (left) ideals and $$S$$-relative submodules are given and also it is shown that $$S$$ is contained in the set theoretic complement of the rejective set of that $$S$$-relative (left) ideal (respectively, submodule). Finally, some of the properties of the relative homomorphisms of $$R$$-modules, and the rejective (respectively, non-absorptive) sets of the group homomorphisms of $$S$$-modules are investigated.

MSC:

 20F05 Generators, relations, and presentations of groups 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)