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ring theory (idea)
See all of ring theory
, no other writeups in this node.
(
idea
)
by
Noether
Mon Jul 03 2000 at 14:56:36
Def:
(
Math
) This branch of
algebra
is the
study
of
rings
.
There are quite a few related
node
s on
everything
.
Basic definitions:
ring
ring homomorphism
subring
quotient ring
centre
,
centralizer
ideal
,
maximal ideal
,
prime ideal
module
,
tensor product
Types of rings:
division ring
field
,
subfield
,
field extension
,
finite field
,
field of fractions
,
characteristic
,
Frobenius endomorphism
integral domain
unique factorization domain
principal ideal domain
Euclidean ring
Noetherian
Examples of rings:
Quaternions
(see also
octonions
)
Weyl algebra
Gaussian integer
s
polynomial
s,
Laurent polynomial
s
rational functions in one variable
integers modulo n
noncommutative geometry
generators and relations for algebras
Properties of elements in rings
multiplicative group
,
unit
associate
prime
irreducible
divides
infinite divisor chain
Theorems about rings
Wedderburn's theorem about finite division rings
,
proof of Wedderburn's theorem about finite division rings
proof that the multiplicative group of a finite field is cyclic
All finite fields are isomorphic to GF(p^n)
invertible matrix theorem
Galois theory
The Hilbert Basis Theorem
Chinese Remainder Theorem
/msg me if you see some more
Ring
Noetherian
proof that the multiplicative group of a finite field is cyclic
quotient ring
Galois Theory
Chinese remainder theorem
Gaussian integer
The Hilbert Basis Theorem
Wedderburn's theorem about finite division rings
tensor product
Proof of Wedderburn's theorem about finite division rings
Weyl algebra
All finite fields are isomorphic to GF(p^n)
Noncommutative Geometry
centralizer
subring
division ring
field
maximal ideal
infinite divisor chain
Euclidean ring
integral domain
finite field
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