- K: any field,
- Z: the ring of integers,
- K: rings of polynomials in one variable with coefficients in a field. (The converse is also true; that is, if A is a PID, then A is a field.) Furthermore, a ring of formal power series in one variable over a field is a PID since every ideal is of the form .
- Z: the ring of Gaussian integers
- Z (where ω is a primitive cube root of 1): the Eisenstein integers
Examples of integral domains that are not PIDs:
- Z: the ring of all polynomials with integer coefficients --- it is not principal because the ideal generated by 2 and X is an example of an ideal that cannot be generated by a single polynomial.
- K: The ideal (x,y) is not principal.
Read more about this topic: Principal Ideal Domain
Other articles related to "example, examples":
... For example, an acronym is a word formed from the initial letter or letters of each of the successive parts or major parts of a compound term (as radar) ... In the examples ananym and metanym, the correct forms (anonym and metonym) were pre-occupied by other meanings ... Other, late 20th century examples, such as hypernym and characternym, are typically incorrectly formed neologisms for which there are more traditional words formed in -onym (hyperonym ...
... Tatira has given a number of examples of proverbs used in advertising in Zimbabwe ... However, unlike the examples given above in English, all of which are anti-proverbs, Tatira's examples are standard proverbs ... proverbs above are meant to make a potential customer smile, in one of the Zimbabwean examples "both the content of the proverb and the fact that it is phrased as a proverb secure the ...
... though /oʊ/ as in toe (other examples dough) tough /ʌf/ as in cuff (other examples rough, enough) cough /ɒf/ as in off (other examples Gough (name, some ...
Famous quotes containing the word examples:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (16881744)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (18961966)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (15331592)