Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Radical of a ring
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
==Definitions== In the theory of radicals, rings are usually assumed to be [[associative]], but need not be [[commutative]] and [[Rng (algebra)|need not have a multiplicative identity]]. In particular, every ideal in a ring is also a ring. <!-- The following definition is adapted mostly from enyclopediaofmaths.org, which is the first reference in the "Further Reading section" --> Let <math>\mathfrak{A}</math> be a class of rings which is: # closed under [[ring homomorphism|homomorphic]] [[image (mathematics)|images]]. That is, for all rings <math>A, B \in \mathfrak{A}</math> and any ring homomorphism <math>f: A \rightarrow B</math> (which may fail to preserve any left- or right-identities) then the image of <math>f</math> is in <math>\mathfrak{A}</math> # closed under taking ideals (for all rings <math>A \in \mathfrak{A}</math>, and <math>I</math> is an ideal on <math>A</math>, then <math>I \in \mathfrak{A}</math>). In particular, <math>\mathfrak{A}</math> could just be the class of all (non-unital) rings. Let ''r'' be some abstract property of rings in <math>\mathfrak{A}</math>. A ring with property ''r'' is called an ''r''-ring; an ideal of some ring with property ''r'' is called an ''r''-ideal. In particular, the ''r''-ideals are a subset of the ''r''-rings. A ring <math>A</math> is said to be a ''r''-'''semi-simple ring''' if it has no non-zero r-ideals. ''r'' is said to be a '''radical property''' if: # the class of ''r''-rings is closed under homomorphic images # For every ring <math>A \in \mathfrak{A}</math> there exists an associated ''r''-ideal <math>r(A)</math>, which is maximal β <math>r(A)</math> contains all the ''r''-ideals of A. The ideal <math>r(A)</math> is called the ''r''- '''radical of the ring''' <math>A</math>. # <math>r( A / r(A) ) = 0</math>, which is true iff the quotient ring <math>A / r(A)</math> is ''r''-semi-simple. Note that, for any ''r''-ring <math>R</math>, <math>R</math> is its own maximal ''r''-ideal. One can say that <math>R</math> is a '''radical''', and the class of ''r''-rings is the '''radical class'''. One can define a radical property by specifying a valid radical class as a subclass of <math>\mathfrak{A}</math>: for an ideal I of some arbitrary ring in <math>\mathfrak{A}</math>, I is an <nowiki>''</nowiki>r<nowiki>''</nowiki>-ideal if it is isomorphic to some ring in the radical class. For any class of rings <math>\delta \in \mathfrak{A}</math>, there is a smallest radical class <math>L \delta</math> containing it, called the '''lower radical''' of <math>\delta</math>. The operator ''L'' is called the '''lower radical operator'''. A class of rings is called '''regular''' if every [[zero ideal|non-zero]] ideal of a ring in the class has a non-zero image in the class.{{clarify|date=March 2025}} For every regular class δ of rings, there is a largest radical class ''U''δ, called the upper radical of δ, having zero intersection with δ. The operator ''U'' is called the '''upper radical operator'''. A radical property ''r'' is said to be '''hereditary''' if for any ring <math>A \in \mathfrak{A}</math> and any ideal <math>I</math> of ring <math>A</math>, <math>r(I)=r(A) \cap I</math>. An equivalent condition on the radical class is that any ideal of a radical is also a radical. The definition readily extends to defining the radical of an [[Algebra (ring theory)|algebra]]. In particular, rings are algebras over the ring of integers.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)