On radical extensions of rings

1967 ◽  
Vol 7 (4) ◽  
pp. 552-554 ◽  
Author(s):  
Efraim P. Armendariz
Keyword(s):  
Jacobson Radical ◽  
Main Tool ◽  
Independent Interest ◽  
Minimum Condition ◽  
Radical Extension ◽  
Commutativity Theorems

A ring K is a radical extension of a subring B if for each x ∈ K there is aninteger n = n(x) > 0 such that xn ∈ B. In [2] and [3], C. Faith considered radical extensions in connection with commutativity questions, as well as the generation of rings. In this paper additional commutativity theorems are established, and rings with right minimum condition are examined. The main tool is Theorem 1.1 which relates the Jacobson radical of K to that of B, and is of independent interest in itself. The author is indebted to the referee for his helpful suggestions, in particular for the strengthening of Theorem 2.1.

Noetherian Tensor Products

1972 ◽  
Vol 15 (2) ◽  
pp. 235-238
Author(s):  
E. A. Magarian ◽  
J. L. Motto
Keyword(s):  
Tensor Product ◽  
Jacobson Radical ◽  
Tensor Products ◽  
Minimum Condition ◽  
Ideal Structure ◽  
The Ideal

Relatively little is known about the ideal structure of A⊗RA' when A and A' are R-algebras. In [4, p. 460], Curtis and Reiner gave conditions that imply certain tensor products are semi-simple with minimum condition. Herstein considered when the tensor product has zero Jacobson radical in [6, p. 43]. Jacobson [7, p. 114] studied tensor products with no two-sided ideals, and Rosenberg and Zelinsky investigated semi-primary tensor products in [9].All rings considered in this paper are assumed to be commutative with identity. Furthermore, R will always denote a field.

scholarly journals Covering and tiling hypergraphs with tight cycles

2020 ◽  
pp. 1-42
Author(s):  
Jie Han ◽  
Allan Lo ◽  
Nicolás Sanhueza-Matamala
Keyword(s):  
Main Tool ◽  
Independent Interest ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Vertex Disjoint

Abstract A k-uniform tight cycle $C_s^k$ is a hypergraph on s > k vertices with a cyclic ordering such that every k consecutive vertices under this ordering form an edge. The pair (k, s) is admissible if gcd (k, s) = 1 or k / gcd (k,s) is even. We prove that if $s \ge 2{k^2}$ and H is a k-uniform hypergraph with minimum codegree at least (1/2 + o(1))|V(H)|, then every vertex is covered by a copy of $C_s^k$ . The bound is asymptotically sharp if (k, s) is admissible. Our main tool allows us to arbitrarily rearrange the order in which a tight path wraps around a complete k-partite k-uniform hypergraph, which may be of independent interest. For hypergraphs F and H, a perfect F-tiling in H is a spanning collection of vertex-disjoint copies of F. For $k \ge 3$ , there are currently only a handful of known F-tiling results when F is k-uniform but not k-partite. If s ≢ 0 mod k, then $C_s^k$ is not k-partite. Here we prove an F-tiling result for a family of non-k-partite k-uniform hypergraphs F. Namely, for $s \ge 5{k^2}$ , every k-uniform hypergraph H with minimum codegree at least (1/2 + 1/(2s) + o(1))|V(H)| has a perfect $C_s^k$ -tiling. Moreover, the bound is asymptotically sharp if k is even and (k, s) is admissible.

scholarly journals Cyclic Flats of a Polymatroid

2020 ◽  
Vol 24 (4) ◽  
pp. 637-648
Author(s):  
Laszlo Csirmaz
Keyword(s):  
Main Tool ◽  
Rank Function ◽  
Independent Interest ◽  
Discrete Measure

Abstract Polymatroids can be considered as “fractional matroids” where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to polymatroids. Defining cyclic flats of a polymatroid carefully, the characterization by Bonin and de Mier of the ranked lattice of cyclic flats carries over to polymatroids. The main tool, which might be of independent interest, is a convolution-like method which creates a polymatroid from a ranked lattice and a discrete measure. Examples show the ease of using the convolution technique.

scholarly journals Compressions and Probably Intersecting Families

2012 ◽  
Vol 21 (1-2) ◽  
pp. 301-313 ◽  
Author(s):  
PAUL A. RUSSELL
Keyword(s):  
Main Tool ◽  
Independent Interest ◽  
Compression Method ◽  
Simple Fact ◽  
New Family ◽  
Fixed Probability ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Nested Sequence

A family of sets is said to be intersecting if A ∩ B ≠ ∅ for all A, B ∈ . It is a well-known and simple fact that an intersecting family of subsets of [n] = {1, 2, . . ., n} can contain at most 2n−1 sets. Katona, Katona and Katona ask the following question. Suppose instead ⊂ [n] satisfies || = 2n−1 + i for some fixed i > 0. Create a new family p by choosing each member of independently with some fixed probability p. How do we choose to maximize the probability that p is intersecting? They conjecture that there is a nested sequence of optimal families for i = 1, 2,. . ., 2n−1. In this paper, we show that the families [n](≥r) = {A ⊂ [n]: |A| ≥ r} are optimal for the appropriate values of i, thereby proving the conjecture for this sequence of values. Moreover, we show that for intermediate values of i there exist optimal families lying between those we have found. It turns out that the optimal families we find simultaneously maximize the number of intersecting subfamilies of each possible order.Standard compression techniques appear inadequate to solve the problem as they do not preserve intersection properties of subfamilies. Instead, our main tool is a novel compression method, together with a way of ‘compressing subfamilies’, which may be of independent interest.

scholarly journals On the Dimension of Modules and Algebras, VI. Comparison of Global and Algebra Dimension

1957 ◽  
Vol 11 ◽  
pp. 61-65 ◽  
Author(s):  
Maurice Auslander
Keyword(s):  
Jacobson Radical ◽  
Minimum Condition ◽  
And Algebra ◽  
Algebra Over A Field

Throughout this paper all rings are assumed to have unit elements. A ring Λ is said to be semi-primary if its Jacobson radical N is nilpotent and Г = Λ/N satisfies the minimum condition. The main objective of this paper isTHEOREM I. Let A be a semi-primary algebra over a field K. Let N be the radical of Λ and Г = Λ/N. IfThendim Λ = gl.dim Λ.

scholarly journals On the Multi-Colored Ramsey Numbers of Paths and Even Cycles

10.37236/5663 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Gábor N. Sárközy
Keyword(s):  
Upper Bound ◽  
Main Tool ◽  
Independent Interest ◽  
Ramsey Numbers

In this paper we improve the upper bound on the multi-color Ramsey numbers of paths and even cycles. More precisely, we prove the following. For every $r\geq 2$ there exists an $n_0=n_0(r)$ such that for $n\geq n_0$ we have $$R_r(P_n) \leq \left( r - \frac{r}{16r^3+1} \right) n.$$ For every $r\geq 2$ and even $n$ we have $$R_r(C_n) \leq \left( r - \frac{r}{16r^3+1} \right) n + o(n) \text{ as }n\rightarrow \infty.$$ The main tool is a stability version of the Erdős-Gallai theorem that may be of independent interest.

Multi-parameter Singular Integrals. (AM-189)

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Partial Differential Equations ◽  
Graduate Students ◽  
General Theory ◽  
Differential Equations ◽  
Singular Integrals ◽  
Main Tool ◽  
Classical Theorem ◽  
Quantitative Version ◽  
Linear Partial Differential Equations

This book develops a new theory of multi-parameter singular integrals associated with Carnot–Carathéodory balls. The book first details the classical theory of Calderón–Zygmund singular integrals and applications to linear partial differential equations. It then outlines the theory of multi-parameter Carnot–Carathéodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. The book then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. This book will interest graduate students and researchers working in singular integrals and related fields.

Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

2013 ◽  
Author(s):  
Friedhelm Waldhausen ◽  
Bjørn Jahren ◽  
John Rognes
Keyword(s):  
Main Tool ◽  
Homotopy Equivalence ◽  
Classifying Space ◽  
Deformation Retract ◽  
Simplicial Sets ◽  
Full Proof

Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.

The Novel TEM Sample Preparation Approach for Targeted via with Barrier/Cu Seed Layer Inspection

2009 ◽  
Author(s):  
Wen-Fei Hsieh ◽  
Shih-Hsiang Tseng ◽  
Bo Min She
Keyword(s):  
Sample Preparation ◽  
Cross Section ◽  
Seed Layer ◽  
Target Location ◽  
Process Development ◽  
Process Integration ◽  
Ion Beam ◽  
Main Tool ◽  
Dual Beam ◽  
Sample Preparation Procedure

Abstract In this study, an FIB-based cross section TEM sample preparation procedure for targeted via with barrier/Cu seed layer is introduced. The dual beam FIB with electron beam for target location and Ga ion beam for sample milling is the main tool for the targeted via with barrier/Cu seed layer inspection. With the help of the FIB operation and epoxy layer protection, ta cross section TEM sample at a targeted via with barrier/Cu seed layer could be made. Subsequent TEM inspection is used to verify the quality of the structure. This approach was used in the Cu process integration performance monitor. All these TEM results are very helpful in process development and yield improvement.

To the discussion about the modern world order

2020 ◽  
pp. 35-41
Author(s):  
A. Mustafabeyli
Keyword(s):  
Soviet Union ◽  
Intergovernmental Relations ◽  
World System ◽  
Main Tool ◽  
World Order ◽  
Modern World ◽  
The Soviet Union ◽  
The World ◽  
And Behavior ◽  
Second World

In many political researches there if a conclusion that the world system which was founded after the Second world war is destroyed of chaos. But the world system couldn`t work while the two opposite systems — socialist and capitalist were in hard confrontation. After collapse of the Soviet Union and the European socialist community the nature of intergovernmental relations and behavior of the international community did not change. The power always was and still is the main tool of international communication.

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