Rings| Rings |
In General > s.a. Semiring.
$ Def: An abelian group
X together with a multiplication, (X, +, · ),
satisfying
– Associativity,
(xy) z = x (yz) for all x,
y, z ∈ X, and
– Distributivity on both
sides, (x+y) z = xz + yz
and x (y+z) = xy + xz.
* More structure: The
multiplication may have an identity (ring with identity), an identity and
inverses, it may be commutative, and commutative with inverses (> see
Field); It may also have a scalar multiplication
(> see algebra).
* Examples: (Smooth) Functions
on a manifold (has an identity); Endomorphisms End(A, A)
of an abelian group A.
@ References: Jacobson 43,
56;
Herstein 69;
Kaplansky 72;
Snaith 03;
Reis 11 [II].
> Online resources: see
Wikipedia page.
Specific Concepts > s.a. Nilpotent Element.
$ Principal ideal:
An ideal in a ring R generated by one element a,
i.e., one of the form Ra.
$ Unit: An element of
a ring which has an inverse.
$ Zero divisor: An element
a in a ring R such that ∃ b ∈ R
with ab = 0.
Commutative
> s.a. Lambda Ring; types of posets.
* Spectrum: For a commutative
ring with identity R, Spec(R) is the set of prime ideals.
* Principal ideal domain:
A commutative ring without divisors of zero in which every ideal is principal,
i.e., a domain where all ideals are principal, or a princial ideal ring without
zero divisors; For example, \(\mathbb Z\).
@ References: Matsumura 87.
Other Types
* Burnside ring of a group G:
Given a finite group G, consider the semiring of isomorphism classes
of finite G-sets (sets on which G acts on the left), with
disjoint union and Cartesian product as operations; The Burnside ring is the
result of applying the Grothendieck construction to this semiring.
* Ordered ring: A ring (A,
+, · ) with a partial order ≤ on the underlying set A that
is compatible with the ring operations in the sense that (i) if x ≤
y then for all z, x + z ≤ y +
z, and (ii) if 0 ≤ x and 0 ≤ y, then 0 ≤
x · y.
* Principal ideal ring: A ring
with identity in which every ideal is principal.
@ Ordered ring: Brumfiel 79 [and semi-algebraic geometry];
> s.a. Wikipedia page.
Ring of Subsets of a Set
$ Def: A collection \(\cal R\)
of subsets of a set X such that for all A, B ∈
\(\cal R\), A \ B ∈ \(\cal R\) and A ∪
B ∈ \(\cal R\).
* Sigma-field / ring:
A ring \(\cal R\) of subsets of a set X, including X,
which is closed under countable unions; X is then called a
measurable space; Example: The σ-field generated by (the
open sets in) a topology, it is called Borel σ-field;
> s.a. Sigma-Algebra.
> Online resources:
see PlanetMath page;
Wikipedia page.
Ring Space
$ Def: A topological space
with a sheaf of rings on it.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 4 nov 2020