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Chiastic structure, or chiastic pattern, is a literary technique in narrative motifs and other textual passages. An example of chiastic structure would be two ideas, A and B, together with variants A' and B', being presented as A,B,B',A'. Chiastic structures that involve more components are sometimes called "ring structures" or "ring compositions".
Chiasmus was particularly popular in the literature of the ancient world, including Hebrew, Greek, Latin and K'iche' Maya, [7] where it was used to articulate the balance of order within the text. Many long and complex chiasmi have been found in Shakespeare [ 8 ] and the Greek and Hebrew texts of the Bible . [ 9 ]
As an example, the nilradical of a ring, the set of all nilpotent elements, is not necessarily an ideal unless the ring is commutative. Specifically, the set of all nilpotent elements in the ring of all n × n matrices over a division ring never forms an ideal, irrespective of the division ring chosen. There are, however, analogues of the ...
A left zero divisor of a ring R is an element a in the ring such that there exists a nonzero element b of R such that ab = 0. [d] A right zero divisor is defined similarly. A nilpotent element is an element a such that a n = 0 for some n > 0. One example of a nilpotent element is a nilpotent matrix.
No nilpotent element can be a unit (except in the trivial ring, which has only a single element 0 = 1). All nilpotent elements are zero divisors . An n × n {\displaystyle n\times n} matrix A {\displaystyle A} with entries from a field is nilpotent if and only if its characteristic polynomial is t n {\displaystyle t^{n}} .
If an element of a ring is nilpotent and central, then it is a member of the ring's Jacobson radical. [15] This is because the principal right ideal generated by that element consists of quasiregular (in fact, nilpotent) elements only. If an element, r, of a ring is idempotent, it cannot be a member of the ring's Jacobson radical. [16]
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All rings are rngs. A simple example of a rng that is not a ring is given by the even integers with the ordinary addition and multiplication of integers. Another example is given by the set of all 3-by-3 real matrices whose bottom row is zero. Both of these examples are instances of the general fact that every (one- or two-sided) ideal is a rng.