Missouri Journal of Mathematical Sciences

Relative Algebraic Structures

Amir M. Rahimi

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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 $\vert G \vert$ the order of the group $G$ is finite, $\vert R(H) \vert = (k-s) \vert H \vert$ 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 $R$-modules are investigated.

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Missouri J. Math. Sci., Volume 14, Issue 2 (2002), 123-135.

First available in Project Euclid: 4 October 2019

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Rahimi, Amir M. Relative Algebraic Structures. Missouri J. Math. Sci. 14 (2002), no. 2, 123--135. doi:10.35834/2002/1402123. https://projecteuclid.org/euclid.mjms/1570154517

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