Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.
A ring is an abelian group (R,+) together with an associative operation * which is distributive over +. Some rings, referred to as unital or unit rings , also have an identity element 1 with respect to *. The operation + is referred to as the addition and * is referred to as the multiplication. The identity element with respect to + is written as 0.
The ring with just one element is called the trivial ring.
Characteristic
The characteristic of a unit ring is the smallest positive integer n satisfying n1=0 if it exists and 0 otherwise. In particular ne=0 for all elements e of the ring.
(It can also be defined as the unique n∈N such that nZ is the kernel of the morphism k -> k 1, from Z to R, i.e. k 1 = 0 <=> k ∈ nZ.)
Types of elements
Central
An element r of a ring R is central if xr = rx for all x in R. The set of all central elements forms a subring of R, known as the center of R.
An element x of a ring is irreducible if for any elements a and b such that x=ab, either a or b is a unit. Note that every irreducible is prime, but not necessarily vice versa.
An element r of the ring R is a unit if there exists an element r-1 such that rr-1=r-1r=1. This element r-1 is uniquely determined by r and is called the multiplicative inverse of r. The set of units forms a group under multiplication.
A nonzero element r of R is said to be a zero divisor if there exists s ≠ 0 such that sr=0 or rs=0. If a ring has a Zero divisor which is also a unit, then the ring has no other elements and is the trivial ring.
Given a ring R and an ideal I of R, the factor ring is the set R/I of cosets {a+I : a∈R} together with operations (a+I)+(b+I)=(a+b)+I and (a+I)*(b+I)=ab+I. The relationship between ideals, homomorphisms, and factor rings is summed up in the fundamental theorem on homomorphisms.
Finitely generated ideal
A left ideal I is finitely generated if there exist finitely many elements a1,...,an such that I = Ra1 + ... + Ran. A right ideal I is finitely generated if there exist finitely many elements a1,...,an such that I = a1R + ... + anR. A two-sided ideal I is finitely generated if there exist finitely many elements a1,...,an such that I = Ra1R + ... + RanR.
A left idealI of R is a subgroup or (R,+) such that aI ⊆ I for all a∈R. A right ideal is a subgroup of (R,+) such that Ia⊆I for all a∈R. An ideal (sometimes for emphasis: a two-sided ideal) is a subgroup which is both a left ideal and a right ideal.
A left ideal of the ring R which is not contained in any other left ideal but R itself is called a maximal left ideal. Maximal right ideals are defined similarly. In commutative rings, there is no difference, and one speaks simply of maximal ideals.
The set of all nilpotent elements in a commutative ring forms an ideal, the nilradical of the ring. The nilradical is equal to the intersection of all the Prime Ideals. It is 'not' equal, in general, to the Jacobson Radical.
An ideal P in a commutative ringR is prime if P ≠ R and if for all a and b in R with ab in P, we have a in P or b in P. Every maximal ideal in a commutative ring is prime. There is also a definition of prime ideal for noncommutative rings.
a principal left ideal in the ring R is a left ideal of the form Ra for some element a of R; a principal right ideal is a right ideal of the form aR for some element a of R; a principal ideal is a two-sided ideal of the form RaR for some element a of R.
The radical of an ideal I in a commutative ring consists of all those ring elements a power of which lies in I. It is equal to the intersection of all maximal ideals containing I.
A ring homomorphism that is bijective is a ring isomorphism. The inverse of an isomorphism, it turns out, is also a ring isomorphism. Two rings are isomorphic if there exists a ring isomorphism between them. Isomorphic rings can be thought as essentially the same, only with different labels on the individual elements.
A ring satisfying the descending chain condition for left ideals is left artinian; if it satisfies the descending chain condition for right ideals, it is right artinian; if it is both left and right artinian, it is called artinian. Artinian rings are noetherian.
An integral domain in which a degree function is defined so that "division with remainder" can be carried out is called a Euclidean domain (because the Euclidean algorithm works in these rings). All Euclidean domains are principal ideal domains.
A commutative division ring is a field. Every finite division ring is a field, as is every finite integral domain. Field theory is in fact an older branch of mathematics than ring theory.
A ring with a unique maximal left ideal is a local ring. These rings also have a unique maximal right ideal, and the left and the right unique maximal ideals coincide.
A ring satisfying the ascending chain condition for left ideals is left noetherian; a ring satisfying the ascending chain condition for right ideals is right noetherian; a ring that is both left and right noetherian is noetherian. A ring is left noetherian if and only if all its left ideals are finitely generated; analogously for right noetherian rings.
A technique to turn a given set of elements of a ring into units. It is named Localization because it can be used to make any given ring into a local ring. To localize a ring R, take a multiplicatively closed subset S containing no zero devisors, and formally define their inverses, which shall be added into R.
A subset S of the ring (R,*,+) which remains a ring when + and * are restricted to S and contains the multiplicative identity 1 of R is called a subring of R.
A rig is an algebraic structure satisfying the same properties as a ring, except that addition need only be an abelian monoid operation, rather than an abelian group. The term "rig" is meant to suggest that it is a "ring" without "negatives".
A rng is an algebraic structure satisfying the same properties as a ring, except that multiplication need not have an identity element. The term "rng" is meant to suggest that it is a "ring" without an "identity".
