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### Shah, Kishor

- Integral Closure of Noetherian Domains and Intersections of Rees Valuation Rings, (I)

#### Authors

#### Source

The Journal of the Indian Mathematical Society, Vol 84, No 1-2 (2017), Pagination: 43-54#### Abstract

It is shown that the integral closure R' of a local (Noetherian) domain R is equal to the intersection of the Rees valuation rings of all proper ideals in R of the form (b, I_{k})R, where b is an arbitrary nonzero nonunit in R and the I

_{k}are an arbitrary descending sequence of ideals (varying with b and with I

_{k}⊆ (I

_{k-1}∩ I

_{1}

^{k}) for all k > 1, one sequence for each b). Also, this continues to hold when b is restricted to being irreducible and no two distinct b are associates. We prove similar results for a Noetherian domain.

#### Keywords

Integral Closure, Noetherian Domain, Local Domain, Rees Valuation Ring.#### References

- M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, MA 1969.
- W. J. Heinzer, L. J. Ratli , Jr., and D. E. Rush, Bases of ideals and Rees valuation rings, J. Algebra 323 (2010), 839-853.
- I. N. Herstein, Topics in Algebra, Cisdell Publishing Co., New York, 1964.
- I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.
- H. Matsumura, Commutative Algebra, W. A. Benjamin, NY, 1970.
- M. Nagata, Local Rings, Interscience, John Wiley, New York, 1962.
- L. J. Ratli , Jr., Note on analytically unrami ed semi-local rings, Proc. Amer. Math. Soc. 17 (1966), 274-279.
- L. J. Ratli , Jr., On prime divisors of the integral closure of a principal ideal, J. Reine Angew. Math. 255 (1972), 210-220.
- D. Rees, Valuations associated with ideals (II), J. London Math. Soc. 36 (1956), 221-228.
- I. Swanson and C. Huneke, Integral Closure of Ideals, Rings and Modules, Cambridge Univ. Press, Cambridge, 2006.

- Integral Closure of Noetherian Domains and Intersections of Rees Valuation Rings, (II)

#### Authors

#### Source

The Journal of the Indian Mathematical Society, Vol 84, No 1-2 (2017), Pagination: 55-72#### Abstract

Let 1 < s_{1} < . . . < s_{k} be integers, and assume that κ ≥ 2 (so s_{k} ≤ 3). Then there exists a local UFD (Unique Factorization Domain) (R,M) such that:

(1) Height(M) = s_{k}.

(2) R = R' = ∩{VI (V,N) € V_{j}}, where V_{j} (j = 1, . . . , κ) is the set of all of the Rees valuation rings (V,N) of the M-primary ideals such that trd((V I N) I (R I M)) = s_{j} - 1.

(3) With V_{1}, . . . , V_{κ} as in (2), V_{1} ∪ . . . V_{κ}is a disjoint union of all of the Rees valuation rings of allof the M-primary ideals, and each M-primary ideal has at least one Rees valuation ring in each V_{j} .

#### Keywords

Integral Closure, Local Domain, Rees Valuation Ring, Unique Factorization Domain.#### References

- M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, MA 1969.
- R. C. Heitmann, Characterization of completions of unique factorization domains, Trans. Amer. Math. Soc. 337 (1993), 379-387.
- I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.
- D. Katz and J. Validashti, Multiplicities and Rees valuations, Collect. Math. 61 (2010), 1-24.
- P. Kemp, L. J. Ratli , Jr., and K. Shah, Integral Closure of Noetherian Domains and Inter- sections of Rees Valuation Rings, (I), J. Indian Math. Soc. (to appear).
- H. Matsumura, Commutative Algebra, W. A. Benjamin, NY, 1970.
- M. Nagata, Local Rings, Interscience, John Wiley, New York, 1962.
- D. G. Northcott, Ideal Theory, Cambridge Tracts in Math. No. 42, Cambridge, 1965.
- L. J. Ratli , Jr., On quasi-unmixed local domains, the altitude formula, and the chain con- dition for prime ideals (II), Amer. J. Math. 92 (1970), 99-144.
- L. J. Ratli , Jr., On prime divisors of the integral closure of a principal ideal, J. Reine Angew. Math. 255 (1972), 210-220.
- I. Swanson and C. Huneke, Integral Closure of Ideals, Rings and Modules, Cambridge Univ. Press, Cambridge, 2006.
- O. Zariski and P. Samuel, Commutative Algebra, Vol. 2, D. Van Nostrand, New York, 1960.

- On Nagata's Result about Height One Maximal Ideals and Depth One Minimal Prime Ideals (I)

#### Authors

**Affiliations**

1 Department of Mathematics, Missouri State University, Springeld, Missouri - 65897, US

2 Department of Mathematics, University of California, Riverside, California - 92521, US

3 Department of Mathematics, Missouri State University, Springfield, Missouri - 65897, US

#### Source

The Journal of the Indian Mathematical Society, Vol 85, No 3-4 (2018), Pagination: 356-376#### Abstract

It is shown that, for all local rings (R,M), there is a canonical bijection between the set *DO(R)* of depth one minimal prime ideals ω in the completion * ^{^}R* of

*R*and the set

*HO(R/Z)*of height one maximal ideals

*̅M'*in the integral closure

*(R/Z)'*of

*R/Z*, where

*Z*:

*= Rad(R)*. Moreover, for the finite sets

**D**:= {

*V*/V**:

*= (*, ω ∈ DO(R)} and H := {

^{^}R/ω)'*V/V := (R/Z)'*}:

_{̅M'},*̅M'*∈ HO(R/Z)(a) The elements in **D** and **H** are discrete Noetherian valuation rings.

(b) **D** = {* ^{^}V* ∈

**H**}.

#### Keywords

Integral Closure, Completion of a Local Ring, Depth One Minimal Prime Ideal, Height One Maximal Ideal.#### References

- Paula Kemp, Louis J. Ratli, Jr., and Kishor Shah, On Nagata's Result About Height One Maximal Ideals and Depth One Minimal Prime Ideals (II), in preparation.
- M. Nagata, Local Rings, Interscience, John Wiley, New York, 1962.
- D. G. Northcott, Ideal Theory, Cambridge Tracts in Math. No. 42, Cambridge, 1965.
- L. J. Ratliff, Jr., On prime divisors of the integral closure of a principal ideal, J. Reine Angew. Math., 255 (1972), 210-220.
- O. Zariski and P. Samuel, Commutative Algebra, Vol. 2, D. Van Nostrand, New York, 1960.

- On Nagata’s Result about Height One Maximal Ideals and Depth One Minimal Prime Ideals (II)

#### Authors

**Affiliations**

1 Department of Mathematics, Missouri State University, Springfield, US

2 Department of Mathematics, University of California, Riverside, US

#### Source

The Journal of the Indian Mathematical Society, Vol 86, No 1-2 (2019), Pagination: 46-57#### Abstract

We expand the theory of height one maximal ideals and depth one minimal prime ideals initiated by M. Nagata and continued by the authors in part I. A local ring is doho in case its completion has at least one depth one minimal prime ideal. We establish several families of doho local rings, prove that certain local rings associated with Rees valuation rings are doho, and complement a famous construction of Nagata by proving that each doho local domain (<I>R,M</I>) of altitude α ≥ 2 has a quadratic integral extension over-domain with precisely two maximal ideals, one of height α and the other of height one.#### Keywords

Completion of a Local Ring, Depth One Minimal Prime Ideal, Height One Maximal Ideal, Rees Valuation Ring.#### References

- Paula Kemp, Louis J. Ratliff, Jr., and Kishor Shah, On Nagata’s result about height one maximal ideals and depth one minimal prime ideals (I), J. Indian Math. Soc., (to appear).
- D. Katz and J. Validashti, Multiplicities and Rees valuations, Collect. Math. 61 (2010), 1-24.
- M. Nagata, Local Rings, Interscience, John Wiley, New York, 1962.
- L. J. Ratliff, Jr., On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals (II), Amer. J. Math. 92 (1970), 99-144.
- I. Swanson and C. Huneke, Integral Closure of Ideals, Rings and Modules, Cambridge Univ. Press, Cambridge, 2006.

- Local Nullstellensatz over Commutative Ground Rings

#### Authors

**Affiliations**

1 Department of Mathematics, Missouri State University, Springfield, Missouri 65897, US

2 Department of Mathematics, University of California, Riverside, California 92521-0135, US

#### Source

The Journal of the Indian Mathematical Society, Vol 90, No 1-2 (2023), Pagination: 149-158#### Abstract

It is shown that a local Nullstellensatz holds over an arbitrary commutative ring A (with identity 1 ≠ 0); specifically, if B = A[x_{1}, . . . , x_{n}] is a finitely generated extension ring of A and N is a maximal ideal in B, then NB_{N} = (N ∩ A, x_{1} − c_{1}, . . . , x_{n} − c_{n})B_{N} for some c_{1}, . . . , c_{n} ∈ B_{N} .

#### Keywords

G-Ideal, Nullstellensatz, Maximal Ideal, Polynomial Ring.#### References

- D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, New York, 1995.
- I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.
- P. Kemp, L. J. Ratliff, Jr., and K. Shah, Depth one homogeneous prime ideals in polynomial rings over a field, Journal of Indian Math. Soc. (accepted).
- M. Nagata, Local Rings, Interscience, John Wiley, New York, 1962.
- O. Zariski and P. Samuel, Commutative Algebra, Vol. 1, D. Van Nostrand, New York, 1958.
- O. Zariski and P. Samuel, Commutative Algebra, Vol. 2, D. Van Nostrand, New York, 1960.

- Depth One Homogeneous Prime Ideals in Polynomial Rings over a Field

#### Authors

**Affiliations**

1 Department of Mathematics, Missouri State University, Springfield, Missouri 65897, US

2 Department of Mathematics, University of California, Riverside, California 92521-0135, US

#### Source

The Journal of the Indian Mathematical Society, Vol 90, No 3-4 (2023), Pagination: 213–232#### Abstract

This paper concerns the question: Which depth one homogeneous prime ideals N in a polynomial ring H are of the principal class? In answer to this question, we introduce acceptable bases of ideals in polynomial rings, and then use a known one-to-one correspondence between the ideals N in H := F[X1, . . . , Xn] such that Xn ∉ N and the maximal ideals P in the related polynomial ring G := F[X1/Xn, . . . , Xn−1/Xn] to show that the acceptable bases of the maximal ideals P in G transform to homogeneous bases. This is used to determine several necessary and sufficient conditions for a given depth one homogeneous prime ideal N in H to be an ideal of the principal class, thus answering, in part, our main question. Then it is shown that the Groebner-grevlex bases of ideals are acceptable bases. Finally, we construct several examples to illustrate our results, and we delve deeper into an example first studied by Macaulay.#### Keywords

Ideal Basis, Polynomial Ring, Prime Ideal.#### References

- W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Math., 39, Cambridge Univ. Press, First paperback ed. with revisions, 1998.
- R. C. Cowsik and M. V. Nori, On the fibres of blowing up, J. Indian Math. Soc. 40(1976), 217-222.
- David Cox, John Little, Donal O’Shea, Ideals, Varieties, and Algorithms, 4th Ed., Springer, New York 2015.
- N. Jacobson, Lectures in Abstract Algebra, Vol. III, D. Van Nostrand, New York, 1964.
- I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.
- M. Nagata, Local Rings, Interscience, John Wiley, New York, 1962.