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, zX, 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
$ 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 ∃ bR 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 inthe sense that (i) if x y then for all z, x + zy + 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 AB ∈ \(\cal R\).
* 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.


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