Understanding Strongly Nilpotent Elements

  • Thread starter peteryellow
  • Start date
In summary, a strongly nilpotent element in a ring R is defined as an element a for which every sequence a_0, a_1, ..., a_i, ... with a_0 = a and a_{i+1} in a_iRa_i, there exists an n with a_n = 0. The author of the conversation is discussing a theorem they are using where a is not strongly nilpotent. This means that there exists a sequence a_0, a_1, ..., a_i, ... with a_{n+1} in a_nRa_n and a_n not equal to zero, where a_1 is also in aRa. This follows from the definition by taking i = 0.
  • #1
peteryellow
47
0
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?
 
Physics news on Phys.org
  • #2
Doesn't that follow immediately from the definition (take i=0)?
 

1. What is the definition of a strongly nilpotent element?

A strongly nilpotent element is an element in a ring or algebraic structure that, when raised to a power, becomes 0 after a finite number of steps.

2. How is a strongly nilpotent element different from a nilpotent element?

A nilpotent element is an element that becomes 0 after a certain number of multiplications, while a strongly nilpotent element becomes 0 after a certain number of powers.

3. Can a strongly nilpotent element be non-zero?

Yes, a strongly nilpotent element can be non-zero. In fact, all non-zero elements in a finite ring are strongly nilpotent.

4. What is the significance of strongly nilpotent elements in algebraic structures?

Strongly nilpotent elements play a significant role in the structure and properties of algebraic structures, particularly in the study of finite rings and algebras. They also have applications in coding theory and other areas of mathematics.

5. How are strongly nilpotent elements related to other algebraic properties?

Strongly nilpotent elements are closely related to other algebraic properties such as idempotency and nilpotency. In fact, an element can be strongly nilpotent if and only if it is both idempotent and nilpotent.

Similar threads

MHB Proving the Formula for Determinants by Induction
    • Linear and Abstract Algebra
Replies
4
Views
958
I How is uniqueness about the determinant proved by this theorem?
    • Linear and Abstract Algebra
Replies
10
Views
2K
MHB Solving Nilpotent Problem in Abstract Algebra
    • Linear and Abstract Algebra
Replies
3
Views
1K
I On a bound on the norm of a matrix with a simple pole
    • Linear and Abstract Algebra
Replies
8
Views
757
MHB Set of nilpotent elements
    • Linear and Abstract Algebra
Replies
4
Views
2K
I Is an algorithm for a proof required to halt?
    • Set Theory, Logic, Probability, Statistics
Replies
3
Views
1K
MHB Principal Ideal Rings and GCDs .... .... Bland Proposition 4.3.3
    • Linear and Abstract Algebra
Replies
1
Views
976
MHB Ring Direct Products .... Bland Problem 3(a), Problem Set 2.1, Page 49 ....
    • Linear and Abstract Algebra
Replies
8
Views
1K
I 3-dimensional split-complex numbers
    • Linear and Abstract Algebra
Replies
1
Views
746
B Is it invalid to redefine the sgn function in this way in a proof?
    • Linear and Abstract Algebra
Replies
15
Views
4K

Hot Threads

Recent Insights

Back
Top