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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".
In rhetoric, chiasmus (/ k aɪ ˈ æ z m ə s / ky-AZ-məs) or, less commonly, chiasm (Latin term from Greek χίασμα chiásma, "crossing", from the Greek χιάζω, chiázō, "to shape like the letter Χ"), is a "reversal of grammatical structures in successive phrases or clauses – but no repetition of words".
Such a ring is necessarily a reduced ring, [5] and this is sometimes included in the definition. In general, if A is a Noetherian ring whose localizations at maximal ideals are all domains, then A is a finite product of domains. [6] In particular if A is a Noetherian, normal ring, then the domains in the product are integrally closed domains. [7]
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.
Also apophthegm. A terse, pithy saying, akin to a proverb, maxim, or aphorism. aposiopesis A rhetorical device in which speech is broken off abruptly and the sentence is left unfinished. apostrophe A figure of speech in which a speaker breaks off from addressing the audience (e.g., in a play) and directs speech to a third party such as an opposing litigant or some other individual, sometimes ...
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 ...
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It is the set of all elements of R that "annihilate" S (the elements for which S is a torsion set). Subsets of right modules may be used as well, after the modification of "sr = 0" in the definition. The annihilator of a single element x is usually written Ann R (x) instead of Ann R ({x}).