View source for Prime ring

==English==
{{wikipedia}}

===Noun===
{{en-noun}}

# {{senseid|en|productofideals}} {{lb|en|algebra|[[ring theory]]}} Any nonzero [[ring]] ''R'' such that for any two (two-sided) [[ideal]]s ''P'' and ''Q'' in ''R'', the product ''PQ'' = 0 (the ''zero ideal'') if and only if ''P'' = 0 or ''Q'' = 0.
#* {{quote-journal|en|year=1969|journal=Taita Journal of Mathematics|volume=1-2|pageurl=https://books.google.com.au/books?id=kNPxAAAAMAAJ&q=%22prime+ring%22%7C%22prime+rings%22&dq=%22prime+ring%22%7C%22prime+rings%22&hl=en&sa=X&ved=0ahUKEwiTwOrl8d7aAhWpv1QKHXvsAT8Q6AEIYTAL|page=56
|passage=A ring is called a '''prime ring''' if the product of nonzero ideals in it remains nonzero. It is obvious that a '''prime ring''' is necessarily semi-prime.}}
#* {{quote-book|en|year=1987|author=Gregory Karpilovsky|title=The Algebraic Structure of Crossed Products|publisher=Elsevier (North-Holland)|url=https://books.google.com.au/books?id=mCwxdPFBT_MC&pg=PA223&dq=%22prime+ring%22%7C%22prime+rings%22&hl=en&sa=X&ved=0ahUKEwiTwOrl8d7aAhWpv1QKHXvsAT8Q6AEI0wEwIg#v=onepage&q=%22prime%20ring%22%7C%22prime%20rings%22&f=false|page=223
|passage=The ring R is said to be ''prime'' if for all nonzero ideals ''A, B'' of ''R'' we have ''AB''≠0. An ideal ''P'' of ''R'' is called a prime ideal if ''R''/''P'' is a '''prime ring'''. '''Prime rings''' and prime ideals are important building blocks in noncommutative ring theory.}}
#* {{quote-book|en|year=2014|author=Matej Brešar|title=Introduction to Noncommutative Algebra|pageurl=https://books.google.com.au/books?id=PkvPBAAAQBAJ&pg=PA163&dq=%22prime+ring%22%7C%22prime+rings%22&hl=en&sa=X&ved=0ahUKEwiTwOrl8d7aAhWpv1QKHXvsAT8Q6AEIwAEwHw#v=onepage&q=%22prime%20ring%22%7C%22prime%20rings%22&f=false|page=163|publisher=Springer
|passage=The so-called extended centroid of a '''prime ring''', i.e., a field defined as the center of the Martindale ring of quotients, will enable us to extend a part of the theory of central simple algebras to general '''prime rings'''.}}
# {{lb|en|algebra|[[ring theory]]}} {{synonym of|en|prime subring}}
#* {{quote-book|en|year=2012|author=Sebastian Xambo-Descamps|title=Block Error-Correcting Codes: A Computational Primer|publisher=Springer Science & Business Media|pageurl=https://books.google.co.uk/books?id=W1MOBwAAQBAJ&pg=PA110&dq=%22the+prime+ring+of%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwijvbPjpbmFAxX9WkEAHap-DLsQuwV6BAgKEAc#v=onepage&q=%22the%20prime%20ring%20of%22&f=false|page=110
|passage=Moreover, the image of φ<sub>A</sub> is the smallest subring of A, in the sense that it is contained in any subring of A, and it is called the '''prime ring''' of A.}}
## {{senseid|en|ownprimesubring}} {{lb|en|algebra|[[ring theory]]||uncommon}} A [[ring]] which is equal to its own [[prime subring]].
##* {{quote-book|en|year=2012|author=Thomas Becker|author2=Volker Weispfenning|title=Gröbner Bases: A Computational Approach to Commutative Algebra|publisher=Springer Science & Business Media|pageurl=https://books.google.co.uk/books?id=tkPtBwAAQBAJ&pg=PA50&dq=%22the+prime+ring+of%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiWqNuJprmFAxWhQEEAHbSNBSgQuwV6BAgLEAg#v=onepage&q=%22the%20prime%20ring%20of%22&f=false|page=50
|passage=The image of ℤ or ℤ/mℤ, respectively, in R as described in the above proposition obviously consists of all sums n · 1 in R, where n ∈ ℤ. It is also called the prime ring of R. R itself is called a '''prime ring''' if it equals its own prime ring. If p is a prime number, then the field ℤ/pℤ is a also called the prime field of characteristic p.}}
##* {{quote-book|en|year=2014|author=Benjamin Fine|author2=Anthony M. Gaglione|author3=Gerhard Rosenberger|title=Introduction to Abstract Algebra: From Rings, Numbers, Groups, and Fields to Polynomials and Galois Theory|publisher=JHU Press|pageurl=https://books.google.co.uk/books?id=zfjZAwAAQBAJ&pg=PA81&dq=%22prime+ring%22+%22isomorphic+to%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwj19YfqsLmFAxVzT0EAHQ1VAD8QuwV6BAgIEAk#v=onepage&q=%22prime%20ring%22%20%22isomorphic%20to%22&f=false|page=81
|passage=Theorem 3.6.3. If R is a '''prime ring''' of characteristic zero then R is isomorphic to ℤ. If R is a '''prime ring''' of characteristic n then R is isomorphic to ℤ<sub>n</sub>.}}

====Usage notes====
The following conditions are equivalent to ''R'' being a '''prime ring''' ({{senseno|en|productofideals}}):
* for arbitrary ''a, b'' ∈ ''R'', if ''arb'' = 0 for all ''r'' ∈ ''R'' (i.e., if ''aRb'' = 0) then either ''a'' = 0 or ''b'' = 0;
* the ''zero ideal'' is a [[prime ideal]] in ''R''.

{{senseno|en|productofideals|uc=1}} and {{senseno|en|ownprimesubring}} are not equivalent; for example, <math>\mathbb{Z}[x]</math> is a prime ring in the sense of {{senseno|en|productofideals}}, but not in the sense of {{senseno|en|ownprimesubring}}.

===See also===
* {{l|en|annihilator}}
* {{l|en|prime ideal}}
* {{l|en|semiprime ring}}

===Further reading===
* {{pedia|Semiprime ring}}
* [https://www.encyclopediaofmath.org/index.php/Prime_ring Prime ring] on {{w|Encyclopedia of Mathematics}}

===Anagrams===
* {{anagrams|en|a=egiimnprr|repriming}}