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**

$ __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

*

*

@

**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.

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jan
2016