Understanding Strongly Nilpotent Elements

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SUMMARY

The discussion centers on the definition and implications of strongly nilpotent elements in ring theory. A strongly nilpotent element 'a' in a ring 'R' is defined such that for any sequence starting with 'a', if the sequence continues with elements in the form 'a_iRa_i', there exists an 'n' such that 'a_n = 0'. The author clarifies that if 'a' is not strongly nilpotent, it implies the existence of a sequence where 'a_n' remains non-zero, specifically noting that 'a_1' belongs to 'aRa'. The confusion arises regarding the derivation of 'a_1 in aRa', which is directly inferred from the definition when 'i=0'.

PREREQUISITES
  • Understanding of ring theory and its terminology
  • Familiarity with sequences and their properties in algebra
  • Knowledge of nilpotent elements and their characteristics
  • Basic grasp of algebraic structures and operations
NEXT STEPS
  • Study the properties of nilpotent elements in algebraic structures
  • Explore the implications of non-nilpotent elements in ring theory
  • Learn about sequences in rings and their convergence properties
  • Investigate related concepts such as Jacobson radical and its applications
USEFUL FOR

Mathematicians, algebraists, and students studying advanced ring theory, particularly those interested in the properties of nilpotent elements and their applications in abstract algebra.

peteryellow
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My definition of a strongly nilpotent element is:

Let a be in ring R the element a is strongly nilpotent if for every sequence a_0,a_1,...,a_i,... such that a_0 =a
and a_{i+1} is in a_iRa_i, there exists an n with a_n =0.


The question is in a theroem I am using that a is not strongly nilpotent, what does it mean

The author is saying

Since a is not strongly nilpotent we have a sequence a_0,a_1,...,a_i,... with a_{n+1} is in a_nRa_n, a_n is different from zero an
and a_1 is in aRa.

I don't understand that why is a_1 is in aRa. how do we get that?
 
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Doesn't that follow immediately from the definition (take i=0)?
 

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