Summary

Advances in algebra have long been an essential part of the development of modern mathematics. This is exemplified by the work of mathematician Norman Steenrod, who made notable contributions to the field of algebraic topology. By studying the properties of homology and cohomology, Steenrod was able to classify different types of topological spaces, which helped advance our understanding of geometric objects. To aid in this work, he also developed the Steenrod algebra, which provided an invaluable tool for investigating the structure of spaces. Steenrod's work has continued to have a significant impact on algebraic topology, and many mathematicians still study his findings today. This is a prime example of the importance of abstraction and generalization in mathematics. By creating broad categories of objects and discovering connections between seemingly unrelated structures, mathematicians can make breakthroughs in multiple fields of study. Another noteworthy advance in algebra is the concept of Noetherian regular rings. A Noetherian regular ring is a type of ring where every element is equal to the product of three other elements. This concept is fundamental in the study of matrices, with the full matrix Mn being regular if and only if N is a regular ring. This idea is extended further to additively commutative semi-Noetherian regular δ-near rings, where the sets of matrices are also rings with new properties. The study of such rings helps in investigating the properties of matrices and their applications in diverse areas of mathematics. In a more practical sense, even scientific publishers require upgrades and maintenance, as demonstrated by World Scientific's upcoming system upgrade. The company, on October 25th, 2022, will undergo an upgrade at 2am EDT, which may cause e-commerce and registration services to be temporarily unavailable for up to 12 hours. However, existing users can still access content during this period, and the company advises those looking to make online purchases to visit their site again after the upgrade is complete. This underlines the importance of ongoing maintenance and updates to ensure that online systems continue to function correctly. In conclusion, advances in algebra have long played a crucial role in the advancement of mathematics, spanning from the abstract concepts of Noetherian regular rings to their practical applications. The exploration of these concepts continues to lead to discoveries and new connections between different areas of mathematics, highlighting the profound importance of algebra in mathematics as a whole.

Consensus Meter

A Noetherian ring N is called a Noetherian regular Ring if every x N, x = xyx for some y N. It is clear understanding that for any ring N and any positive integer n, the full matrix ring Mn is regular if and only if N is a regular ring. For a positive integer n and an additively commutative semiNoetherian Regular δ-Near Ring S with zero, let Mn(S) be the set of all n x n matrices over S. Then under the usual addition and multiplication of matrices, Mn(S) is also an additively commutative semi- Noetherian Regular δ-Near Ring with zero and then n x n zero matrix over S is the zero of matrix semi- Noetherian Regular δ-Near Ring Mn(S). Definition 1.4: A Commutative ring N with identity is a Noetherian Regular δ-Near Ring if it is Semi Prime in which every non-unit is a zero divisor and the Zero ideal ia s Product of a finite number of principle ideals generated by semi prime elements and N is left simple which has N0 = N, Ne = N. Definition 1.5: A triple is called a semi-Noetherian Regular δ-Near Ring if and are Semigroups and is distributive over. Then the Noetherian Regular δ-Near ring Mn(S) is Noetherian Regular δ-Near ring if and only if S is a Noetherian Regular δ-Near ring.

Published By:

N Nagar - Advances in Algebra, 2011 - researchgate.net

Cited By:

35

This article discusses the work of mathematician Norman Steenrod in the field of algebraic topology. Steenrod made major contributions to the study of homology and cohomology, which help to classify different types of topological spaces. He also developed the Steenrod algebra, a powerful tool for studying the structure of spaces. Steenrod's work had a profound impact on the field of algebraic topology and continues to be studied today. The article concludes by noting that Steenrod's work is an example of the importance of abstraction and generalization in mathematics, which allows mathematicians to study broad classes of objects and discover new connections between seemingly unrelated structures.

Published By:

S MacLane - The American Mathematical Monthly, 1939 - Taylor & Francis

Cited By:

21

On October 25th, 2022, World Scientific's system will undergo an upgrade at 2am EDT. Existing users will still be able to access content, but e-commerce and registration for new users might not be available for up to 12 hours. The company advises people to visit their site again for online purchases and to reach out to customer care for any concerns. In summary, World Scientific's website will undergo maintenance on October 25th, 2022, which may affect e-commerce and registration for up to 12 hours. Current users will still be able to access existing content, and the company advises people to visit their site again for online purchases after the upgrade has completed.

Published By:

KB Nam - Advances In Algebra, 2003 - World Scientific

Cited By:

27

The proceedings volume of the Southern Regional Algebra Conference (SRAC) held in March 2017 covers a range of research topics in algebra. The papers presented in the volume include both theoretical and computational methods, and cover areas such as ring theory, group theory, commutative algebra, algebraic geometry, linear algebra, and quantum groups. The papers consist of research articles and survey papers and highlight ongoing research in algebraic geometry, combinatorial commutative algebra, computational methods for representations of groups and algebras, Lie superalgebras, and tropical algebraic geometry. SRAC has been held since 1988, and this volume showcases the latest findings in computational and theoretical methods in algebra and representation theory. The book is suitable for graduate students and researchers interested in algebraic research.

Published By:

J Feldvoss, L Grimley, D Lewis, A Pavelescu, C Pillen - Springer

Cited By:

0

The article discusses the general notion of independence introduced by E. Marczewski in 1958. A set I of the carrier A of an algebra is called M-independent if equality of two term operations f and g of the algebra on any finite system of different elements of I implies f = g in A. While there are interesting results on this notion of independence, it is not wide enough to cover stochastic independence, independence in projective spaces, and some others. Hence, weaker notions of independence such as Q-independence, which relies on families Q of mappings into A, have been developed. The article delves into Q-independence and discusses the Galois correspondence between families Q of mappings and Q-independent sets. The article concludes with some easily formulated and interesting results. The article is dedicated to the memory of Professor B.H. Neumann.

Published By:

K Głazek - Advances In Algebra, 2003 - World Scientific

Cited By:

6

From the activities and sub questions that were analysed, all 533 activities in the FPRWs and all 196 activities in the ANAs could be classified into four types of patterns: number patterns, repeating patterns, shape patterns with growth and other patterns that did not fit into the three categories. The categories were labelled Difficulty Level 1, 2 and 3, respectively - abbreviated to DL1, DL2 and DL3. DL1 repeating patterns are those patterns where the core of the pattern is interrupted, because these activities simply require the drawing of the next item(s) in the collection of shapes. Although DL2 patterns do not explicitly employ the notion of core to extend patterns, the given core is not interrupted, as it is in DL1. DL3 repeating patterns are complex patterns, where the core of the pattern is fully shown and has two or more items with multi-variability. These patterns can be classified according to levels of cognitive engagement, which range from simple patterns to more complex patterns where the core of the pattern is not easily recognised.

Published By:

J Du Plessis - South African Journal of Childhood Education, 2018 - journals.co.za

Cited By:

15

The World Scientific website will undergo a system upgrade on October 25th, 2022, at 2 am (EDT). Existing users will be able to log in and access content, but registration of new users and e-commerce may be unavailable for up to 12 hours. Visitors are advised to check back later for online purchases. Any inquiries can be directed to customercare@wspc.com. In conclusion, World Scientific is informing users of its website that it will be down for an upgrade on October 25th, 2022, from 2 am (EDT). Customers are urged to be aware that new user registration and e-commerce services may be unavailable for up to 12 hours. Existing users, however, will be able to log in and view content. For additional information or inquiries, please contact customercare@wspc.com.

Published By:

SK Jain, P Kanwar, JB Srivastava - Advances In Algebra, 2003 - World Scientific

Cited By:

6