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In mathematics, specifically in [[abstract algebra]], a '''prime element''' of a [[commutative ring]] is an object satisfying certain properties similar to the [[prime number]]s in the integers and to [[irreducible polynomial]]s. Care should be taken to distinguish prime elements from [[irreducible element]]s, a similar concept which is the same in many rings.
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==Definition==
An element <math>p</math> of a [[commutative ring]] <math>R</math> is said to be '''prime''' if it is not zero or a [[unit (ring theory)|unit]] and whenever <math>p</math> [[Divisibility_(ring_theory)|divides]] <math>ab</math> for some <math>a</math> and <math>b</math> in <math>R</math>, then <math>p</math> divides <math>a</math> or <math>p</math> divides <math>b</math>.  Equivalently, an element <math>p</math> is prime if, and only if, the [[principal ideal]] <math>(p)</math> generated by <math>p</math> is a nonzero [[prime ideal]].<ref>{{harvnb|Hungerford|1980|loc=Theorem III.3.4(i)}}, as indicated in the remark below the theorem and the proof, the result holds in full generality.</ref>
 
Interest in prime elements comes from the [[Fundamental theorem of arithmetic]], which asserts that each [[integer]] can be written in essentially only one way as 1 or −1 multiplied by a product of positive [[prime number]]s. This led to the study of [[unique factorization domain]]s, which generalize what was just illustrated in the integers.
 
Being prime is relative to which ring an element is considered to be in; for example, 2 is a prime element in '''Z''' but it is not in '''Z'''[<math>i</math>], the ring of [[Gaussian integers]], since <math>2=(1+i)(1-i)</math> and 2 does not divide any factor on the right.
 
==Connection with prime ideals==
{{Main|Prime ideal}}
An ideal ''I'' in the ring ''R'' (with unity) is [[prime ideal|prime]] if the factor ring ''R''/''I'' is an [[integral domain]].
 
A nonzero [[principal ideal]] is [[prime ideal|prime]] if and only if it is generated by a prime element.
 
==Irreducible elements==
{{Main|Irreducible element}}
Prime elements should not be confused with [[irreducible element]]s.  In an [[integral domain]], every prime is irreducible<ref>{{harvnb|Hungerford|1980|loc=Theorem III.3.4(iii)}}</ref> but the converse is not true in general. However, in unique factorization domains,<ref>{{harvnb|Hungerford|1980|loc=Remark after Definition III.3.5}}</ref> or more generally in [[GCD domain]]s, primes and irreducibles are the same.
 
==Examples==
The following are examples of prime elements in rings:
* The integers ±2, ±3, ±5, ±7, ±11,... in the [[ring of integers]] '''Z'''
* the complex numbers  (<math>1+i</math>), 19, and (<math>2+3i</math>) in the ring of [[Gaussian integers]] '''Z'''[<math>i</math>]
* the polynomials <math>x^2 - 2</math> and <math>x^2+1</math> in the [[ring of polynomials]] over '''Z'''.
 
==References==
;Notes
{{reflist}}
 
;Sources
*Section III.3 of {{Citation
| authorlink=Thomas W. Hungerford
| last=Hungerford
| first=Thomas W.
| title=Algebra
| edition=Reprint of 1974
| year=1980
| publisher=[[Springer-Verlag]]
| location=New York
| series=Graduate Texts in Mathematics
| volume=73
| isbn=978-0-387-90518-1
| mr=0600654
}}
*{{citation  |author=Jacobson, Nathan  |title=Basic algebra. II  |edition=2  |publisher=W. H. Freeman and Company  |place=New York  |year=1989  |pages=xviii+686  |isbn=0-7167-1933-9  |mr=1009787}}
 
*{{citation  |author=Kaplansky, Irving  |title=Commutative rings  |publisher=Allyn and Bacon Inc.   |place=Boston, Mass.  |year=1970  |pages=x+180  |mr=0254021 }}
 
[[Category:Ring theory]]

Latest revision as of 19:33, 30 December 2014

The mommy kitty looked at me from inside the cage at the local animal shelter as if she were asking, "What did I do wrong?"
Poor kitty! You did nothing wrong. This world is simply cruel, especially to homeless cats that have the misfortune of giving birth to kittens.

I managed to trap two feral cats, both of which had kittens. Total: 2 adult female cats and 11 kittens. They would have been food for coyotes, and their numbers could have run into the hundreds if more females had survived long enough to reproduce.

But these cats did nothing wrong. Their only crime was being born without humans to love them, care for them and get them spayed.
I was feeding them in an attempt to catch them, but I failed on every attempt to lure them into the cage with food. The kittens, on the other paw, proved to be a sufficient lure.

The first mommy cat went nuts after I shut the door, trapping her inside the cage with her kittens, and she never calmed down. She will probably be euthanized as unadoptable. The second mommy cat was tame enough for me to pick her up and put her inside the cage with her kittens, but she faces a similar fate.

Most on the spot trans 7 kecelakaan terbesar di dunia people adopt kittens, not adult cats.
And how are they going to find homes for 11 kittens? They try their best, but that is a big order.
Those kitties did nothing wrong! I feel so bad!
My own cats, all spayed and neutered, cuddled with me all night. I still feel so bad! I wish that I could adopt every homeless cat, but I can't.

Cruel world!
MaryannStutchbu (talk)

If you loved this information and you want to receive more info about on the spot trans 7 assure visit our webpage.

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