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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)
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