Radical Ideals Research Articles

Social Reconstruction through Education: The Philosophy, History, and Curricula of a Radical Ideal

Is Social Transformation Always Progressive? Rightist Reconstructions of Schooling Today, Michael W. Apple The Social Frontier 1934-1943: Retrospect and Prospect, James R. Giarelli Red Teachers Can't Save Us: Radical Educators and Liberal Journalists in the 1930s, James Wallace Harold Rugg's Social Reconstructionism, Peter F. Carbone and Virginia S. Wilson Female Founders and the Progressive Paradox, Susan F. Semel Southern Progressivism During the Great Depression: Virginia and Black Social Reconstruction, Michael James Social Recon-structionism for 21st Century Educators, William O. Stanley and Kenneth D. Benne

Past Imperfect: French Intellectuals, 1944-1956

The uniquely prominent role of French intellectuals in European cultural and political life following World War II is the focus of Tony Judt's newest book. He analyzes this intellectual community's most divisive conflicts: how to respond to the promise and the betrayal of Communism and how to sustain a commitment to radical ideals when confronting the hypocrisy in Stalin's Soviet Union, in the new Eastern European Communist states, and in France itself. Judt shows why this was an all-consuming moral dilemma to a generation of French men and women, how their responses were conditioned by war and occupation, and how post-war political choices have come to sit uneasily on the conscience of later generations of French intellectuals. Judt's analysis extends beyond the writings of fashionable 'Existentialist' personalities such as Jean-Paul Sartre, Albert Camus, and Simone de Beauvoir to include a wide intellectual community of Catholic philosophers, non-aligned journalists, literary critics and poets, Communist and non-Communist alike. Judt treats the intellectual dilemmas of the postwar years as an unfinished history. French intellectuals have not fully come to terms with the gnawing sense of what Judt calls the 'moral irresponsibility' of those years. The result, he suggests, is a legacy of bad faith and confusion that has damaged France's cultural standing, notably in newly liberated Eastern Europe, and which reflects the nation's larger difficulty in confronting its own ambivalent past.

Finitely Generated Radical Ideals in H ∞

Finitely Generated Radical Ideals in H ∞

Finitely generated radical ideals in 𝐻^{∞}

It is shown that a radical ideal in H ∞ {H^\infty } is finitely generated if and only if it is a principal ideal generated by a Blaschke product having simple zeros.

Further results on reticulated rings

For more details about the reticulation LR of a ring, see Simmons [9]. For any ideal I of the ring R, let D(I) be the ideal generated by {D(a) : a E I ) in LR. For any ideal J of the lattice LR, let D-l (J ) = {a E R: D(a) E J). Trivially, D-I (J ) is an ideal of R. The reticulation of a ring was investigated by Simmons [9] in order to show that a lo t of ring theoretic properties have analogues in lattice theory and vice versa. In this paper we continue this theme and answer a couple of questions raised by Simmons [9]. Then we proceed to prove further results in that direction. Let Id(R) be the lattice of all ideals of the ring R. It should be noted that this lattice is not necessarily distributive. Let RId(R) be the distributive lattice of radical ideals of the ring R. For a lattice L, let Id(L) be the lattice of ideals of L. From now on, LR will denote the reticulation of R. We start by quoting a theorem that was given by Johnstone [7], p. 194.

A general theory of structure spaces with applications to spaces of prime ideals

A structure space is a quadrupleX=(X, d, A, P), where for some setR, X ⊂A=2 R ,d:X×X →A is defined byd(I, J)=J−I, andP is the family of cofinite subsets ofR. Forr e P, I e X, N r (I)={J e X: d(I, J) ⊂r},To(X)={Q ⊂X: if x e Q there is anr e P such thatN r (x) $$ \subseteq$$ Q}. ThenTo(X) is a (not usually Hausdorff) topology onX called the hull-kernel topology. Replacing d byd *, whered * (I, J)=d(J, I), or byd s, whered s (I, J.)=d(I, J) ∪d * (I, J), and proceeding in the obvious way yields thedual hull-kernel topology To(X *) andsymmetric topology To(X s ). The latter is always a zero-dimensional Hausdorff space. When R is a commutative ring with identity andX is a collection of proper prime ideals ofR, To(X s ) is usually called thepatch topology. Our generality enables us to improve on known results in the case of space of prime ideals and to apply this theory to a wide variety of algebraic structures. In particular, we establish criteria for a subspace of a structure space to be closed in the symmetric topology; we establish a duality between families of maximal elements in the hull-kernel topology and families of minimal elements in the dual hull-kernel topology of subspaces that are closed in the symmetric topology; we use topological constructions to generalize certain ring theoretic notions, such as radical ideals an annihilator ideals; we use this theory to obtain new results about subspaces of the space prime ideals of a reduced, commutative ring.

Radical Ideals of Dedekind Domains and their Extensions

The question of whether or not an ideal of a ring can be a radical is studied, and a characterisation of the ideals which can be radical is obtained for Dedekind domains and their polynomial and matrix rings. Some partial results on this question are given for other extension rings of Dedekind domains.

Fuzzy prime ideals and fuzzy radical ideals

Some properties of the fuzzy prime ideals and radical ideals are studied. Also we study the structure of fuzzy principal ideals.

On the radical ideals of seminormal rings

On the radical ideals of seminormal rings

Free resolutions of certain codimension three perfect radical ideals

Introduction. The original conception of this work included a complete study of certain codimension three perfect ideals that arise naturally in a set-up, with the purpose of enriching the present collection of such ideals. The ones we consider are always radical and their syzygy-theoretic properties are relatively easy to describe. From the viewpoint of the theory of graded resolutions, they share common numerical features with (the projecting cones of) certain codimension two and three varieties in projective space. Their resolutions being seldom pure, they seemed largely convenient for testing the sharpness of some calculations made in the pure case (cf. [19], [25], [26]). Moreover, they yield a large class suited for testing recently developed concepts, such as the notion of strong obstruction, that have its natural place in the theory of linkage ([loc. cit.]) or the more general theory of residual intersection developed in [16]. The authors did not pursue this line of thought, but it seemed worthwhile pointing it out. A brief description of the present contents is as follows. In the first section one obtains the explicit free resolution of a large class of codimension three ideals that appear as presentation ideals of normal cones associated to determinantal ideals fixing a submatrix. As it turns out, the resolutions of these presentation ideals are given by a mapping cylinder of a suitable map to the resolution of the original determinantal ideals. The second section is concerned with deriving some ideal-theoretic results from the knowledge of the presentation given in section one. This procedure, by and large, follows the techniques developed by various authors ([17], [14], [23] and [24]). In the third section one gives the resolution of the determinantal ideal Jr associated to a map m ~ R (m > n), fixing r c m, in the case of the data r = n - 2 and m = n + 2. This case not only extends the previous known ones [1], [2], but mainly gives some genuine clue for the general situation. The construction follows closely the spirit of the scandinavian complex [9] in that one is led to tensor suitable complexes and then to cut down the resulting size by an use of trace maps. This method seems to generalize well in various contexts [21]. Last one establishes exact values for the codimension of the complete intersection locus of the ideals considered in the previous sections. The reason for doing so is that the known estimates to present [11], [27] would give no real grasp, in this case, of the locus.

On the lifting problem for homogeneous ideals in polynomial rings

We prove here that if k is a field of zero characteristic, then any homogenous ideal in k[X, Y] is liftable to a radical ideal. On the other hand, if k is a finite field, then for any n ≥ 2, there exist zero-dimensional monomial ideals in k[ X 1,…, X n ] which are not liftable to radical ideals.

Hochster’ theorem, coherent locales, and lattices of radical ideals

(1988). Hochster’ theorem, coherent locales, and lattices of radical ideals. Communications in Algebra: Vol. 16, No. 12, pp. 2625-2648.

Some Connections between Heyting Valued Set Theory and Algebraic Geometry —Prolegomena to Intuitionistic Algebraic Geometry—

Rousseau [10] has shown that classical or standard function theory of n variables is no other than intuitionistic function theory of one variable over \boldsymbol C^{n–1} . Similar works have been done by Nishimura [9] in the realm of Sato hyperfunctions and by Takeuti and Titani [16] in the realm of complex manifolds. The main purpose of this paper is to pursue similar results in the arena of algebraic geometry. Since we would like to do so in an intuitionistically valid manner, we reconstruct some rudiments of algebraic geometry, using the complete Heyting algebra of radical ideals in place of the space of prime ideals with Zariski topology as the starting point of our scheme theory.

Nonfinite unramified extensions

Suppose A is a normal noetherian domain containing Q and C =) A is a domain which is affine and unramified over A. Let B be the integral closure of A in C, the field of quotients of C. Now C is normal and thus A c B c C. This gives rise to four radical ideals, the Finiteness and Discriminant ideals of A, and the Zariski Complement and Branch Locus ideals of B. The purpose of the paper is to analyze these ideals and their behavior under Galois closure and tensor product. Also we give a topological proof of a theorem of Zariski: the purity of the Discriminant ideal. Since B c C is unramified, it follows from Zariski’s Main Theorem that Spec C + Spec B is an open immersion. The complement of its image is V(I) where Zc B is a radical ideal. Define the Zariski Complement ideal by ZC(C, B) = I. Define the Finiteness ideal by Fint(C, A) = {UE A; C, is finite over A,}. The statement that an ideal J of a commutative ring has pure height 1 means that every minimal prime over J has height 1. Then the Zariski Complement has pure height 1 in B, the Finiteness ideal has pure height 1 in A, and ZC(C, B) n A = Fint(C, A). Furthermore, each of these ideals commutes with localization in A. Let n be the dimension of the field extension 2 c B, and V= (U E B; the minimal polynomialf, E A[ r] of u has degree n}. Define the Branch Locus ideal BL(B, A) as the radical in B of the ideal generated by all&(u) where UE V. Let disc(u) be the discriminant of U. Define the Discriminant ideal Disc(B, A) as the radical in A of the ideal generated by all disc(u) where u E V. If p c B is a prime ideal, B is unramilied over A at p iff BL(B, A) d p. If q c A is a prime ideal, B is unramilied over A at every extension of q iff Disc(B, A) ti q. Also BL(B, A) n A = Disc( B, A), Fint(C, A) c Disc(B, A), and ZC(C, B) c BL(B, A). If A is regular, it is a theorem due to Zariski, Nagata, and Auslander that BL(B, A) and Disc(B, A) have pure height 1. Let G(C) be the normalization (or Galois closure) of C over 2, and let t7, )...) 0” . C --* G(C) be the distinct embeddings of C which are the identity on A. If R is a domain with A c R c C, define G(R) to be the ring 254 oool-8708/87 $7.50

Standard bases of perfect homogeneous polynomial ideals of height 2

This paper concerns minimal systems of homogeneous generators (i.e., standard bases) of homogeneous ideals Z in a commutative polynomial ring K with coefficients in a field k. A basic numerical question about a standard basis B of Z is to determine its cardinality or, more generally, the number of the elements in B of degree t, for each t > 0; this number depends only on Z (and not on the choice of B) and will be denoted by v,(Z). In this paper we obtain a sharp estimate for the numbers v,(P), when P is a perfect homogeneous ideal of height 2 in K. Precisely, in (2.1) we provide upper and lower bounds for v,(P) in terms of the Hilbert function H = H(K/P, ) of K/P and of an integer /I, which ranges in an interval defined by H. Then in (3.3) we show the sharpness of these bounds: in fact we prove a more general result by exhibiting examples of ideals P (with given H and /I) having, for each degree t, one of the possible values v,(P) expected from (2.1). Finally, in (4.1) we modify the preceding examples and obtain, under certain hypotheses, radical ideals with the same properties. The preceding results offer refined versions of well known theorems. In fact, from (2.1) we obtain (cf. (2.3)) a more complete form of Dubreil’s inequalities (cf. [2, 11); moreover (3.3) and (4.1) improve, in the particular hypotheses under consideration, a classic result of Macaulay (cf. [lo, 143) and its reduced version (cf. [6, 11, 131). Finally, particular cases of our estimates in (2.1) can be found in [7, 8, 5, 121 (for points “in generic position” in the projective plane), in [ 1 ] (for “Hilbert-function-completeintersections”), and in [4] (for monomial ideals of height 2).

Lattice Aspects of Radical Ideals and Choice Principles

Lattice Aspects of Radical Ideals and Choice Principles

Guilt and Shame in the Women's Movement: The Radical Ideal of Action and Its Meaning for Feminist Intellectuals

Guilt and Shame in the Women's Movement: The Radical Ideal of Action and Its Meaning for Feminist Intellectuals

Valuation Subalgebras and Analogues of the Decomposition of a Radical Ideal into Prime Ideals

Valuation Subalgebras and Analogues of the Decomposition of a Radical Ideal into Prime Ideals

Further Inspiration for Tiberius Gracchus?

The force of example is nowhere greater than in the adaptation of institutions observed abroad and modified for local conditions. Is it not possible, that in addition to his Stoic mentors, in addition to the radical ideals of Agis and Cleomenes, Tiberius Gracchus may have had the practical inspiration of a contemporary system of land settlement before him? It is my suggestion that Ptolemaic Egypt offered Roman agrarian reforms in the second century B.C. an example worthy of imitation, either in part or whole, and that such nobiles as M. Aemilius Lepidus, Laelius, Aemilianus and, not least, Tiberius Gracchus himself were attracted by the way the Ptolemies ran the rural sector of their economy.

Radical ideals in group rings of locally finite groups

Perhaps the most interesting and difficult problem in the study of the group ring K[ G] is the determination of the Jacobson radical JK[ G] of this ring. To date this has been achieved only in a number of special cases, notably for G solvable or linear. It now appears that this problem splits into two and that a solution of the cases with G finitely generated and with G locally finite would then handle the general case. In this paper we consider group rings of locally finite groups. More specifically, we propose a candidate for JK[ G] here and we show that this candidate is at least a radical ideal.