More Group Theory and Ring Theory

2011 ◽  
pp. 295-328
Keyword(s):  
2014 ◽  
Vol 60 ◽  
pp. 127-150 ◽  
Author(s):  
George Bergman ◽  
Trevor Stuart

Paul Cohn was born in Hamburg, where he lived until he was 15 years of age. However, in 1939, after the rise of the Nazis and the growing persecution of the Jews, his parents, James and Julia Cohn, sent him to England by Kindertransport. They remained behind and Paul never saw them again; they perished in concentration camps. In England, being only 15 years old, he was directed to work first on a chicken farm but later as a fitter in a London factory. His academic talents became clear and he was encouraged by the refugee committee in Dorking and by others to continue his education by studying for the English School Certificate Examinations to sit the Cambridge Entrance Examination. He was awarded an Exhibition to study mathematics at Trinity College. After receiving his PhD in 1951, Paul Cohn went from strength to strength in algebra and not only became a world leader in non-commutative ring theory but also made important contributions to group theory, Lie rings and semigroups. He was much admired, and he travelled widely to collaborate with other algebraists. Moreover, he was a great supporter of the London Mathematical Society, serving as its President from 1982 to 1984.


Author(s):  
Andreas Bolfing

Blockchains are heavily based on mathematical concepts, in particular on algebraic structures. This chapter starts with an introduction to the main aspects in number theory, such as the divisibility of integers, prime numbers and Euler’s totient function. Based on these basics, it follows a very detailed introduction to modern algebra, including group theory, ring theory and field theory. The algebraic main results are then applied to describe the structure of cyclic groups and finite fields, which are needed to construct cryptographic primitives. The chapter closes with an introduction to complexity theory, examining the efficiency of algorithms.


Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 411
Author(s):  
Faisal Mehmood ◽  
Fu-Gui Shi ◽  
Khizar Hayat ◽  
Xiao-Peng Yang

In traditional ring theory, homomorphisms play a vital role in studying the relation between two algebraic structures. Homomorphism is essential for group theory and ring theory, just as continuous functions are important for topology and rigid movements in geometry. In this article, we propose fundamental theorems of homomorphisms of M-hazy rings. We also discuss the relation between M-hazy rings and M-hazy ideals. Some important results of M-hazy ring homomorphisms are studied. In recent years, convexity theory has become a helpful mathematical tool for studying extremum problems. Finally, M-fuzzifying convex spaces are induced by M-hazy rings.


Author(s):  
Carl Faith ◽  
Stanley Page
Keyword(s):  

Author(s):  
Pierre Ramond
Keyword(s):  

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