Amount algebras

In this paper, as a generalization to content algebras, we introduce amount algebras. Similar to the Anderson–Badawi [Formula: see text] conjecture, we prove that under some conditions, the formula [Formula: see text] holds for some amount [Formula: see text]-algebras [Formula: see text] and some ideals [Formula: see text] of [Formula: see text], where [Formula: see text] is the smallest positive integer [Formula: see text] that the ideal [Formula: see text] of [Formula: see text] is [Formula: see text]-absorbing. A corollary to the mentioned formula is that if, for example, [Formula: see text] is a Prüfer domain or a torsion-free valuation ring and [Formula: see text] is a radical ideal of [Formula: see text], then [Formula: see text].

Primeradicals in Ternary Semi Groups

In this paper we study the properties of prime radical of an ideal in a ternarysemigroup. We characterize different classes of ternarysemigroups by their properties of their radicals and nilpotent. We introduced and charaterize the notions of radical ideal generated by P in ternarysemigroups.

Jurisprudence of Love in Paul’s Letter to the Romans

Paul proclaims that Christ is the end of law. The new Christian community promises to be a community sustained not by law, but by love. His zeal for love reinforces his proclamation of Christ as the end of law, for love is lawless, literally outside of law. In contrast to Moses who establishes the rule of law on Mount Sinai, Paul proclaims the power of love over law and asserts the inherent lawlessness of love. Legal rules predict human behaviour; but love makes human actions unpredictable. Legal rules dictate outcomes; but love is, by definition, free. However, inasmuch as love is free, love is also fleeting. According to Paul, Christianity marks the end of law and the dawn of love, but it does not take long for the first church council to be formed and decrees to be issued. The promise of a lawless community ends up with codes of canon law. The modern political state carries with it this ancient theological baggage. We inherit from Paul a particular cognitive dissonance: we dream in the language of love, but speak in the language of law. The radical ideal of love as the ultimate negation of law remains a powerful eschatological vision in our theo-political imaginary.

Bit-size reduction of triangular sets in two and three variables

At ISSAC 2004, Schost and D.introduced a transformation of triangular lexicographic Groebner basesgenerating a radical ideal of dimension zero,which reduces significantly the bit-size of coefficients.The case where the triangular lexicographic Groebner basis does not generate a radical idealis far more complicated. This work treats the case of n=2 variables, andin some extent the case of n=3 variables.It resorts to an extra operation, the squarefree factorization;nevertheless this operation has low complexity cost.But as soon as n>2 variables a lack of simple and efficient gcd-like operationover non-reduced rings prevents to undertake meaningful algorithmic considerations.An implementation in Maple for the case n=2 confirms the expectedreduction of the expected size coefficients.

Symbolic Powers of Ideals Defining F-pure and Strongly F-regular Rings

Given a radical ideal $I$ in a regular ring $R$, the Containment Problem of symbolic and ordinary powers of $I$ consists of determining when the containment $I^{(a)} \subseteq I^b$ holds. By work of Ein–Lazersfeld–Smith, Hochster–Huneke and Ma–Schwede, there is a uniform answer to this question, but the resulting containments are not necessarily best possible. We show that a conjecture of Harbourne holds when $R/I$ is F-pure, and prove tighter containments in the case when $R/I$ is strongly F-regular.

COMMENTS REGARDING d-IDEALS OF CERTAIN f-RINGS

Let A be a reduced commutative f-ring with identity and bounded inversion, and let A* be its subring of bounded elements. By first observing that A is the ring of fractions of A* relative to the subset of A* consisting of elements which are units in the bigger ring, we show that the frames Did (A) and Did (A*) of d-ideals of A and A*, respectively, are isomorphic, and that the isomorphism witnessing this is precisely the restriction of the extension map I ↦ Ie which takes a radical ideal of A* to the ideal it generates in A. Specializing to the ring [Formula: see text], we show that if L is an F-frame, then the saturation quotient of [Formula: see text] is isomorphic to βL. We also investigate projectability properties of [Formula: see text] and [Formula: see text], where the latter denotes the frame of z-ideals of [Formula: see text]. We show that [Formula: see text] is flatly projectable precisely when [Formula: see text] is a feebly Baer ring. Quite easily, [Formula: see text] is projectable if and only if L is basically disconnected. Less obvious is that [Formula: see text] is projectable if and only if L is cozero-complemented.

$q,t$-Catalan Numbers and Generators for the Radical Ideal defining the Diagonal Locus of $({\mathbb C}^2)^n$

Let $I$ be the ideal generated by alternating polynomials in two sets of $n$ variables. Haiman proved that the $q,t$-Catalan number is the Hilbert series of the bi-graded vector space $M(=\bigoplus_{d_1,d_2}M_{d_1,d_2})$ spanned by a minimal set of generators for $I$. In this paper we give simple upper bounds on $\text{dim }M_{d_1, d_2}$ in terms of number of partitions, and find all bi-degrees $(d_1,d_2)$ such that $\dim M_{d_1, d_2}$ achieve the upper bounds. For such bi-degrees, we also find explicit bases for $M_{d_1, d_2}$.

Radical Ideals in Valuation Domains

An idealIof a ringRis called a radical ideal ifI= ℛ(R) where ℛ is a radical in the sense of Kurosh–Amitsur. The main theorem of this paper asserts that ifRis a valuation domain, then a proper idealIofRis a radical ideal if and only ifIis a distinguished ideal ofR(the latter property means that ifJandKare ideals ofRsuch thatJ⊂I⊂Kthen we cannot haveI/J≅K/Ias rings) and that such an ideal is necessarily prime. Examples are exhibited which show that, unlike prime ideals, distinguished ideals are not characterizable in terms of a property of the underlying value group of the valuation domain.

LEFSCHETZ PRINCIPLE APPLIED TO SYMBOLIC POWERS

In this paper, an alternative proof is presented of the following result on symbolic powers due to Ein, Lazarsfeld and Smith [3] (for the affine case over [Formula: see text]) and to Hochster and Huneke [4] (for the general case). Let A be a regular ring containing a field K. Let [Formula: see text] be a radical ideal of A and let h be the maximum of the heights of its minimal primes. Then for all n, we have an inclusion [Formula: see text], where the first ideal denotes the hnth symbolic power of [Formula: see text]. In prime characteristic, this result admits an easy tight closure proof due to Hochster and Huneke. In this paper, the characteristic zero version is obtained from this by an application of the Lefschetz Principle.