Abstract
In this paper we prove that which of Green's relations $\mathcal{L,R,H}$ and $\mathcal{D}$ in rings preserve the minimality of quasi-ideal. By this it is possible to show the structure of the classes generated by the above relations which have a minimal quasi ideal. For the completely simple rings we show that they are generated by the union of zero with a $\mathcal{D} $-class. Also we emphasize that a completely simple ring coincides with the union of zero with a $\mathcal{D} $-class if and only if it is a division ring.
Highlights
By a ring we mean a ring, which does not neccessarily have a identity
This implies that if two elements a, b of a ring A are aHb [aDb], the quasi ideal (a)q is minimal if and only if (b)q is a minimal quasi ideal. After this we show the structure of H- class which contain a minimal quasi ideal,by pointing out that such a H- class is a union of minimal quasi ideals, which as rings are isomorphic with each other
We can only show that a L [R, D] class that contains a minimal quasi-ideal is a union of minimal quasi-ideals and each two of them are isomorphic as additive groups
Summary
By a ring we mean a ring, which does not neccessarily have a identity. Green’s relations are successful tools for studying semigroups. In this paper by using Green’s Lemma for relation L, [R] we prove that for each to elements a, b of a ring A which are aLb [aRb] the quasi ideal (a)q is minimal if and only if (b)q is a minimal quasi ideal.
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