Moroccan Journal of Algebra and Geometry with Applications
Relative prime (resp., semiprime) ideals with applications in C(X)
Alireza Olfati and Ali Taherifar
Department of Mathematics, Yasouj University, Yasouj, Iran
Department of Mathematics, Yasouj University, Yasouj, Iran
Pages 428-441 | Received 20 August 2022, Accepted 01 November 2022, Published 06 December 2022
Abstract
Let I and J be two ideals in a commutative ring R. The ideal I is called J-prime (resp., J-semiprime) if a,b \in J (resp., a \in J) and ab \in I (resp., a^2 \in I) imply a \in I or b \in I (resp., a \in I). Whenever J \nsubseteq I and I is a J-prime (resp., J-semiprime) ideal, the ideal I is said to be a relative prime (resp., semiprime) ideal, and moreover, the ideal J is a p (resp., sp)-factor of I. The class of relative semiprime ideals includes relative z-ideals in any commutative ring and all nonessential ideals in reduced commutative rings. In this article, first we characterize some properties of these two classes of ideals in any commutative ring. Next, we apply the theory of relative prime (resp., semiprime)-ideals in the ring of continuous functions.
Keywords: Relative z-ideal, prime ideal, F-space.
MSC numbers: Primary 13A15; Secondary 54C40.
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