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On coprimely structured rings

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In this paper, we define coprimely structured rings, which are the generalization of strongly 0-dimensional rings. Furthermore, we investigate coprimely structured rings and give some relations between other rings such as Artinian rings, strongly 0 dimensional rings, and h-local domains. | Turk J Math (2016) 40: 719 – 727 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1502-28 Research Article On coprimely structured rings ¨ ˙ IS ˙ ¸ CI˙ 1,∗, K¨ ¨ ˙ 3 Neslihan Ay¸sen OZK IR ur¸sat Hakan ORAL2 , Unsal TEKIR ˙ Department of Mathematics, Faculty of Science and Arts, Yıldız Technical University, Istanbul, Turkey 2 ˙ Department of Mathematics, Faculty of Science and Arts, Yıldız Technical University, Istanbul, Turkey 3 ˙ Department of Mathematics, Faculty of Science and Arts, Marmara University, Istanbul, Turkey 1 Received: 09.02.2015 • Accepted/Published Online: 26.08.2015 • Final Version: 16.06.2016 Abstract: In this paper, we define coprimely structured rings, which are the generalization of strongly 0-dimensional rings. Furthermore, we investigate coprimely structured rings and give some relations between other rings such as Artinian rings, strongly 0-dimensional rings, and h-local domains. Key words: Prime ideal, Artinian ring, coprimely structured ring 1. Introduction Throughout this paper, we assume that R is a commutative ring with identity. A proper ideal P of R is called as a prime ideal if for any a, b ∈ R , ab ∈ P implies a ∈ P or b ∈ P . Moreover, Spec(R) denotes the set of prime ideals of R and M axSpec(R) denotes the set of maximal ideals of R . It is known that the nilradical of R , N , is equal to the intersection of all prime ideals of R . The dimension of R , denoted by dim (R) , is defined to be sup{n ∈ Z+ |there exists a strict chain of prime ideals of R of length n} (see also [11], for more information). Prime ideals are an area of interest in many fields such as algebra and algebraic geometry. The following is the well-known prime avoidance theorem: if an ideal I of a ring R is contained in a union of finitely many prime ideals Pi ’s, then it is contained in Pi for some i . This property was extended to infinite union by Reis and Viswanathan in [10] and they called .