TAILIEUCHUNG - Corrigendum: On density theorems for rings of Krull type with zero divisors

This corrigendum is written to correct some parts of the paper “On density theorems for rings of Krull type with zero divisors”. The proofs of Proposition and Proposition are incorrect and the current note makes the appropriate corrections. | Turk J Math (2017) 41: 1446 – 1447 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Corrigendum: On density theorems for rings of Krull type with zero divisors Ba¸sak AY SAYLAM∗ ˙ ˙ Department of Mathematics, Faculty of Science, Izmir Institute of Technology, Izmir, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: This corrigendum is written to correct some parts of the paper “On density theorems for rings of Krull type with zero divisors”. The proofs of Proposition and Proposition are incorrect and the current note makes the appropriate corrections. Key words: Krull ring, ring of Krull type, additively regular rings In [3], for the Marot ring provided in Example , [2, Proposition ] does not hold. Thus, we change our hypothesis “Marot” to “additively regular” in [2, Proposition ] and reprove it. Proposition Let R be an additively regular ring and P, P1 , . . . , Pn a collection of prime regular ideals ∪n such that P ⊈ Pi for any i . Then Reg(P ) ⊈ i=1 Pi . ∪n Proof We have that P ⊈ Pi for any i , and hence, by [1, Proposition ], P ⊈ i=1 Pi . Thus, ∪n there is a ∈ P − i=1 Pi . Since the product of regular elements is regular, there exists a regular element b ∈ P ∩ P1 ∩ P2 ∩ . . . ∩ Pn . Thus, there exists a u ∈ R such that x = a + ub is regular in R , and ∪n 2 x ∈ Reg(P ) − i=1 Pi . This change we make in [2, Proposition ] affects only [2, Lemma ] and [2, Proposition ], where the hypothesis “Marot” is changed to “additively regular”. Furthermore, we note that the 2-generated regular ideal A , found in [2, Theorem ], may not be invertible. Proposition Let R be an additively regular ring of Krull type. Denote by vi the valuation associated with the valuation ring R(Pi ) , where Pi is the center of vi for each i and by Gi the associated value group. For the 2 -generated regular ideal A ,

• Toán học
• Krull ring
• Ring of Krull type
• Additively regular rings
• Appropriate corrections
• Zero divisors