Comment on "Two notes on imbedded prime divisors"
aa r X i v : . [ m a t h . A C ] M a y Comment on “Two notes onimbedded prime divisors”
Rahul Kumar & Atul Gaur Department of MathematicsUniversity of Delhi, Delhi, India.E-Mail: [email protected]; [email protected]
Abstract
In this note, we show that a part of [5, Remark 2.2] is not correct.Some conditions are given under which the same holds.
Mathematics Subject Classification:
Primary 13E05, Secondary 13B99.
Keywords:
Noetherian rings, Normal pair, Adjacent rings.The following result was proved in [5 , Remark . Theorem 0.1. If R ⊂ T are Noetherian rings such that there does notexist any integrally dependent adjacent Noetherian rings between them, thenfor each ¯ c/ ¯ b ∈ T /Z (where Z = Rad ( T ) = Rad ( R ) and ¯ b, ¯ c regular in R/Z ),we have either ¯ c/ ¯ b ∈ R/Z or ¯ b/ ¯ c ∈ R/Z , and so ( R/Z )[¯ c/ ¯ b ] is a localizationof R/Z . The following examples show that the above result is not correct:
Example 0.2. (1) Let R = Z and T = Z [1 /p ], where p is a prime number.Then clearly R ⊂ T are adjacent Noetherian rings, R is integrally closedin T , and Rad ( R ) = Rad ( T ) = 0. Now, q/p ∈ T \ R for any prime q distinct from p but p/q / ∈ R .(2) Let R = Z (+) M and T = Z [1 /p ](+) M , where p is a prime number and M is any finitely generated R -module as well as T -module. Clearly, R is The author was supported by the SRF grant from UGC India, Sr. No. 2061440976. The author was supported by the MATRICS grant from DST-SERB India, No.MTR/2018/000707.
Rahul Kumar and Atul Gaur integrally closed in T and R ⊂ T is adjacent. Also, by [1, Theorem 4.8], R ⊂ T are Noetherian rings. Note that Z = Rad ( R ) = Rad ( T ) =0(+) M and so by [1, Theorem 3.1], R/Z = Z and T /Z = Z [1 /p ]. Now, q/p ∈ T /Z for any prime q distinct from p but neither q/p / ∈ R/Z nor p/q / ∈ R/Z . Remark 0.3. (1) Let R ⊂ T be an extension of Noetherian rings suchthat there does not exist any integrally dependent adjacent Noetherianrings between them. Then R is integrally closed in T , by the proof of[5, Theorem 2.1], and hence Rad ( T ) = Rad ( R ). In addition, if Z ( T ) = Z ( R ), then Z ( T /Rad ( R )) = Z ( R/Rad ( R )) as the set of zero-divisors ina reduced ring is the union of minimal prime ideals. Now, if Theorem0.1 holds, then it is easy to see that R/Rad ( R ) and T /Rad ( R ) are local(proof follows mutatis mutandis from the proof of [2, Theorem 1.5] and[2, Corollary 1.6]). Consequently, R and T are local. Conversely, if R ⊂ T is an extension of local Noetherian rings such that there does notexist any integrally dependent adjacent Noetherian rings between them,then the proof of [5, Theorem 2.1] shows that Theorem 0.1 holds. Inparticular, for an extension of domains, Theorem 0.1 holds if and only ifthe domains are local.(2) Note that if ( R, T ) is a normal pair (in the sense of [3]), then there doesnot exist any integrally dependent adjacent rings between them. Let R ⊂ T be an extension of Noetherian rings such that ( R, T ) is a normalpair. Then
T /Rad ( R ) is an overring of R/Rad ( R ), by the proof of [5,Theorem 2.1]. Now, if there is at most one maximal ideal of R whichis not minimal prime, then the proof of Theorem 0.1 follows mutatismutandis from the proof of [4, Proposition 3.6(b)] and part (1). Notethat the condition that R is complemented in [4, Proposition 3.6(b)] isnot required here. References [1] D. D. Anderson, M. Winders, Idealization of a module, J. Commut.Algebra, 1(1) (2009) 3-56.[2] A. Ayache, O. Echi, Valuation and pseudovaluation subrings of anintegral domain, Comm. Algebra, 34(7) (2006) 2467-2483.[3] D. E. Davis, Overrings of commutative rings III: Normal pairs, Trans.Amer. Math. Soc. 182 (1973) 175-185.[4] D.E. Dobbs, J. Shapiro, Normal pairs with zero-divisors, J. AlgebraAppl. 10(2) (2011) 335-356. omment on “Two notes on imbedded prime divisors”omment on “Two notes on imbedded prime divisors”