Understanding Strongly Nilpotent Elements

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Strongly nilpotent elements in a ring R are defined such that for any sequence starting with an element a, if each subsequent element is derived from the previous one using the ring's multiplication, the sequence will eventually reach zero. The discussion centers on a theorem stating that if an element a is not strongly nilpotent, there exists a sequence where the elements remain non-zero. A key point of confusion arises regarding why a_1 is in aRa, which seems to follow directly from the definition when considering the initial element a. Clarification is sought on the implications of this definition in the context of the theorem. Understanding these relationships is crucial for grasping the properties of strongly nilpotent elements.
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)?
 
I am studying the mathematical formalism behind non-commutative geometry approach to quantum gravity. I was reading about Hopf algebras and their Drinfeld twist with a specific example of the Moyal-Weyl twist defined as F=exp(-iλ/2θ^(μν)∂_μ⊗∂_ν) where λ is a constant parametar and θ antisymmetric constant tensor. {∂_μ} is the basis of the tangent vector space over the underlying spacetime Now, from my understanding the enveloping algebra which appears in the definition of the Hopf algebra...

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