Topological semirings which are union of topological rings
Keywords:
Topological semiring, topological ring, idempotent semiring, completely regular (algebraic) semiring, normal idealAbstract
In this paper, we establish some necessary and sufficient conditions on a topological completely regular (algebraic) semiring to be a union of topological skew-rings. As a consequence we show that a topological completely regular (algebraic) semiring $(S, \tau)$ satisfying the identity $x+0_y+0_x+y=0_x+y+x+0_y$ is an idempotent semiring of topological rings if and only if for each $G \in \tau$ and every $a \in G$, there exists an element $U \in \tau$ such that $a \in U \subseteq G \cap H_a^{+}$. Also, we prove that the quotient space $S / K$ of a topological completely regular (algebraic) semiring ( $S, \tau$ ) by a normal ideal $K$ is a topological idempotent semiring. Finally, we establish that the quotient space of topological product of a family of topological semirings by product of their normal ideals is topologically isomorphic with the topological product of the family of quotient spaces of topological semiring by its corresponding normal ideal.
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