On the embedded primary components of ideals. IV

Journal of Pure and Applied Algebra, 1995

Communications in Algebra, 1997

Let I be an M -primary ideal in a local ring (R, M ) and let irr(I) denote the set of irreducible components of I, where an ideal q is an irreducible component of I if q occurs as a factor in some decomposition of I as an irredundant intersection of irreducible ideals. We give several characterizations of the ideals in irr(I) and show that if J is an ideal between I and an irreducible component of I, then J is the intersection of ideals in irr(I). We also exhibit examples showing that there may exist irreducible ideals containing I that contain no ideal in irr(I). Also, we determine necessary and sufficient conditions that the principal ideal uR [u, tI] of the Rees ring R[u, tI] have a unique cover, and apply this to the study of the form ring of R with respect to I.

arXiv: Rings and Algebras, 2018

We establish the primary decomposition and uniqueness of primary decomposition for k-ideals in commutative Noetherian semirings.

Formalized Mathematics, 2021

Summary. We formalize in the Mizar System [3], [4], definitions and basic propositions about primary ideals of a commutative ring along with Chapter 4 of [1] and Chapter III of [8]. Additionally other necessary basic ideal operations such as compatibilities taking radical and intersection of finite number of ideals are formalized as well in order to prove theorems relating primary ideals. These basic operations are mainly quoted from Chapter 1 of [1] and compiled as preliminaries in the first half of the article.

Communications in Algebra, 2003

: 13C99, 13H99, 14D25 Let R be a Noetherian ring and I an ideal in R. Then there exists an integer k such that for all n ≥ 1 there exists a primary decomposition Also, for each homogeneous ideal I in a polynomial ring over a field there exists an integer k such that the Castelnuovo-Mumford regularity of I n is bounded above by kn. The regularity part follows from the primary decompositions part, so the heart of this paper is the analysis of the primary decompositions. In [S], this was proved for the primary components of height at most one over the ideal. This paper proves the existence of such a k but does not provide a formula for it. In the paper [SS], Karen E. Smith and myself find explicit k for ordinary and Frobenius powers of monomial ideals in polynomial rings over fields modulo a monomial ideal and also for Frobenius powers of a special ideal first studied by Katzman. Explicit k for the Castelnuovo-Mumford regularity for special ideals is given in the papers by Chandler [C] and Geramita, Gimigliano and Pitteloud [GGP]. Another method for proving the existence of k for primary decompositions of powers of an ideal in Noetherian rings which are locally formally equidimensional and analytically unramified is given in the paper by Heinzer and Swanson [HS]. The primary decomposition result is not valid for all primary decompositions. Here is an example: let I be the ideal (X 2 , XY ) in the polynomial ring k[X, Y ] in two variables X and Y over a field k. For each positive integer m, I = (X) ∩ (X 2 , XY, Y m ) is an irredundant primary decomposition of I. However, for each integer k there exists an integer m, say m = k + 1, such that (X, Y ) k ⊆ (X 2 , XY, Y m ). Hence the result can only

2007

ln this paper, we give some characterizations for prime and primary submodules of a finitely generated free modules over PID's and determine the height of prime submodules. We also characterize the minimal primary decompositions and radicals of submodules of any finitely generated free module over a PID.

2000

We prove that each ideal of a locally formally equidimensional analytically unramied Noetherian integral domain is the contraction of an ideal of a one-dimensional semilocal birational extension domain. We give an application to a problem concerning the primary decomposition of powers of ideals in Noetherian rings. It is shown in an earlier paper by the second author that for each

2012

Let $\mathcal{Z(R)}$ be the set of zero divisor elements of a commutative ring $R$ with identity and $\mathcal{M}$ be the space of minimal prime ideals of $R$ with Zariski topology. An ideal $I$ of $R$ is called strongly dense ideal or briefly $sd$-ideal if $I\subseteq \mathcal{Z(R)}$ and $I$ is contained in no minimal prime ideal. We denote by $R_{K}(\mathcal{M})$, the set of all $a\in R$ for which $\overline{D(a)}= \overline{\mathcal{M}\setminus V(a)}$ is compact. We show that $R$ has property $(A)$ and $\mathcal{M}$ is compact if and only if $R$ has no $sd$-ideal. It is proved that $R_{K}(\mathcal{M})$ is an essential ideal (resp., $sd$-ideal) if and only if $\mathcal{M}$ is an almost locally compact (resp., $\mathcal{M}$ is a locally compact non-compact) space. The intersection of essential minimal prime ideals of a reduced ring $R$ need not be an essential ideal. We find an equivalent condition for which any (resp., any countable) intersection of essential minimal prime ideals ...

Acta Mathematica Academiae Scientiarum Hungaricae, 1976

There are available in the literature many criteria for a ring to have the property that its torsion modules (in the sense of Dickson) decompose into direct sums of primary submodules. The study of rings with this property was inauguarated by S. E. DICKSON in [4] and [5], where such rings were termed T-rings, and has been continued by numerous authors ([1 ], [2], [9], [11 ], [16], et al.). In this note we confine our attention to commutative rings with identity and develop a criterion for such a ring to be a T-ring in terms of the pIime spectrum. The prime spectrum has been used as a tool for investigating torsion of commutative regular rings by C. NLsT3,-SESCU in [10]. What motivated us to find our criterion were several queries by Tom Cheatham (private communication), namely: What commutative regular rings are T-rings ? In particular, is an arbitrary product of fields a T-ring ? The criteria in the above mentioned papers do not seem to be readily applicable to these questions. However, Theorem 1 below yields a fairly straightforward answer to the first question and an atfirmative answer to the second one. In fact, our results show that self-injective regular rings and rings of continuous functions on completely regular spaces are T-rings. In this connection I would like to thank Roger Wiegand for showing me how to prove the latter assertion for compact spaces. The general result needed only a modification of his proof.