# Associated Primes and Double Annihilator Ideals

In this talk I will present a new characterizations of the associated prime ideals of a Noetherian ring or the associated prime ideals of an ideal in a Noetherian ring. Since the terminology associated prime ideal of a commutative ring with unit has in the text books available to me several different definitions, that in the end usually are equivalent, let us agree for the sake of this talk, that an associated prime ideal of a ring $A$ is a prime ideal $p$ $\subset$ $A$ that is the annihilator ideal of a nonzero element of $w$ $\in$ $A$; $w$ is then said to be a witness to $p$ $\subset$ $R$ being an associated prime ideal of $R$. In broad terms we will be concerned with finding a witness for an associated prime ideal and the structure of the set of witnesses.

We begin with a result of F. S. Macaulay [15] $\S$26, which I stumbled across completely by accident between the years 2021 and 2022 as one says in german,$^{1}$ which solves another problem we worked on in the pre-corona days. This result of F. S. Macaulay, like many of his others, seems to have been completely forgotten in the course of time. It concerns the quotient construction for two ideals $(a$ $\underset{R}{:}$ $b)$ in a polynomial ring.$^{2}$ Suitably interpreted it is a sort of precursor of the Noether Involution Theorem [16]. It is in the lovely paper [6] of W. Gröbner which provided the first published proof of E. Noether’s result.

$LEMMA$ $1$ (F. S. Macaulay): Let $R$ be a commutative ring with unit and $a$, $b$ $\subset$ $R$ a pair of ideals in $R$. Then $(a$ $\underset{R}{:}$ $(a$ $\underset{R}{:}$ $(a$ $\underset{R}{:}$ $b)))$ $=$ $(a$ $\underset{R}{:}$ $b)$.

Lemma 1 has a number of remarkable consequences that involve only the special case of this lemma where $a$ $=$ $(0)$ is the zero ideal.

$COROLLARY$ $2$: Let $R$ be a commutative ring with unit and $b$ $\subset$ $R$ an ideal. Then Ann$_{R}($Ann$_{R}($Ann$_{R}$$(b)))$ $=$ Ann$_{R}$$(b)$.

This can be rephrased as the followng double annihilator result for annihilator ideals.

$COROLLARY$ $3$: Let $R$ be a commutative ring with unit and $k$ $\subset$ $R$ an ideal which is the annihilator ideal of an ideal $l$ $\subset$ $R$. Then $k$ $=$ Ann$_{R}($ Ann$_{R}(k))$, so an annihilator ideal is always its own double annihilator.

From Corollary 3 we obtain the following generalization of a result of S. E. Landsburg [12] who proved it for the minimal associated primes of a Noetherian ring.

$THEOREM$ $4$: Let $R$ be a commutative Noetherian ring and $p$ $\subset$ $R$ an associated prime of $R$. Then $p$ $=$ Ann$_{R}($Ann$_{R}(p))$ so $p$ is it’s own double annihilator ideal as is Ann$_{R}(p)$.

A fair number of consequences for associated prime ideals follow from this and are in the manuscript [21] which has been submitted and is waiting for a referee report for almost a year now.

$^{1}$ Meaning between Xmas 2021 and New Years day 2022.

$^{2}$ F. S. Macaulay’s proof was by means of elimination theory according to a remark in [17] and W. Gröbner's proof, by contrast, is in the spirit of E. Noether. Although the result is attributed by W. Gröbner [6] to F.S. Macaulay the proof here is that of W. Gröbner.

$References$

[1] R. Berger, Über verschiedene Differentbegriffe, Sitzungsberichte der Heidelberger Akadamie derWissenschaft.

[2] A. Broer, Differents in Modular Invariant Theory, Trans. Groups 11 (2006), xxx–yyy.

[3] C. McDaniel and L. Smith, Equivariant Coinvariant Rings and Bott–Samelson Rings II: Structure Theorems, Preprint (submitted), AG-Invariantentheorie, 2019.

[4] C. McDaniel, L. Smith, and J. Watanabe, Enveloping Algebras, Preprint (in preparation), AG-Invariantentheorie, 2019.

[5] S. E. Landsburg, Patching Modules of Finite Projective Dimension, Comm. in Algebra, 19:7 (1985), 1461-1473.

[6] W. Gröbner, Irreducible Ideale in kommutativen Ringe, Math. Ann. 110 (1934), 197-222.

[7] H. C. Hutchins, Examples of Commutative Rings, Polygonal Publishing House, Passaic, NJ 1981.

[8] N. Jacobson, Lectures in Abstract Algebra, 3 Vols, D. van Nostrand C$^{o.}$, Princeton, NJ, 1951, 1953,1964.

[9] I. Kaplansky, Commutative Rings, Allyn and Bacon, Inc. Boston, MA 1970.

[10] W. Krull, Idealtheorie, zweite ergänzte Auflage, Erg. Math. Bd. 46, Springer-Verlag, Heidelberg, Berlin1968.

[11] K. Kuhnigk, On Macaulay Duals of Hilbert Ideals, JPAA 210 (2007), 473–480 http://dx.doi.org.10.1016/jpaa.2006.10.14.

[12] S. E. Landsburg, Patching Modules of Finite Projective Dimension, Comm. in Algebra, 19:7 (1985), 1461-1473.

[13] E. Lasker, Zur Theorie der Moduln und Ideals, Math. Ann. 60 (1905), 20-116.

[14] F. S. Macaulay, On the Resolution of a given Modular System into Primary Systems, Math. Ann. 74 (1913), 66–121.

[15] F. S. Macaulay, The Algebraic Theory of Modular Systems, Camb. Math. Lib., Camb. Camb. Univ. Press, Cambridge 1916 (reissued with an introduction by P. Roberts 1994).

[16] D. M. Meyer and L. Smith, Poincaré Duality Algebras, Macaulay’s Dual Systems, and Steenrod Operations, Cambridge University Press, Cambridge, UK, Tracts in Mathematics 167, 2005.

[17] E. Noether, Idealtheorie in Ringbereichen, Math. Ann. 83 (1921), 24–66.

[18] L. Smith, An Application of the Noether Involution Theorem, NonPreprint as T$_{E}$X file, 2019 revised 2021.

[19] L. Smith, A Proof of the Noether Involution Theorem, NonPreprint as a T$_{E}$X file, NIT.TEX, 2021.

[20] L. Smith, Notes on Primary Ideals, NonPreprint as T$_{E}$X file, 2021.

[21] L. Smith, A Note on Double Annihilator Ideals, Preprint, AG-Invariantentheorie, 2022.

[22] O. Zariski and P. Samuel, Commutative Algebra, Volumes I,II, Graduate Texts in Math. 28, 29, Springer-Verlag, Berlin, New York, 1975.