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.
Definition of a ring
A ring is an abelian group (R,+) together with an associative operation * which is distributive over + and has an identity element 1 with respect to *. The operation + is referred as the addition and * is referred 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 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.
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.
- Idempotent
- An element e of a ring is idempotent if e2 = e.
- Irreducible
- An element r of a ring is irreducible if for any elements a and b such that x=a b, either a or b is a unit. Note that every irreducible is prime, but not necessarily vice versa.
- prime
- An element x of a ring is prime if for any elements a and b such that x=a b, either x divides a or x divides b.
- Nilpotent
- An element r of R is nilpotent if there exists a positive integer n such that rn = 0.
- Unit or invertible element
- 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.
- Zero divisor
- 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.
Homomorphisms and ideals
- Factor 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.
- Ideal
- A left ideal I 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.
- Jacobson radical
- The intersection of all maximal left ideals in a ring forms a two-sided ideal, the Jacobson radical of the ring.
- Kernel of a ring homomorphism
- It is the preimage of 0 in the codomain of a ring homomorphism. Every ideal is the kernel of a ring homomorphism and vice versa.
- Maximal 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.
- Nilradical
- 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.
- Prime ideal
- An ideal P in a commutative ring R 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.
- Principal ideal
- 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.
- Radical of an ideal
- 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.
- Ring homomorphism
- A function f : R → S between rings (R,+,*) and (S,⊕,×) is a ring homomorphism if it has the special properties that
- f(a + b) = f(a) ⊕ f(b)
- f(a * b) = f(a) × f(b)
- f(1) = 1
- for any elements a and b of R.
- Ring isomorphism
- A ring homomorphism that is bijective is a ring isomorphism. The inverse of an isomorphism, it turns out, is also a ring homomorphism. 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.
Types of rings
- Artinian ring
- 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. Commutative artinian rings are noetherian.
- Boolean ring
- A ring in which every element is idempotent is a boolean ring.
- Commutative ring
- A ring R is commutative if the multiplication is commutative, i.e. rs=sr for all r,s∈R.
- Division ring or skew field
- A ring in which every nonzero element is a unit and 1≠0 is a division ring.
- Euclidean domain
- 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.
- Field
- A commutative division ring is a field. Every finite division ring is a field, as is every finite integral domain. Field theory is indeed an older branch of mathematics than ring theory.
- Integral domain
- A commutative ring without zero divisors and in which 1≠0 is an integral domain.
- Local ring
- 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.
Any ring can be made local via localization.
- Noetherian ring
- 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 ring that has a "nice" decomposition. A semi-simple ring is also Noetherian, and has no nilpotent Ideals. A ring can be made semi-simple if it is divided by it's Jacobson Radical.
- Simple ring
- A ring with no two-sided Ideals.
- Principal ideal domain
- An integral domain in which every ideal is principal is a principal ideal domain. All principal ideal domains are unique factorization domains.
Miscellaneous
- Direct product and direct sums
- These are ways to construct new rings from given ones; please refer to the corresponding links for explanation.
- Krull dimension of a commutative ring
- The maximal length of a strictly increasing chain of prime ideals in the ring.
- Localization of a ring
- 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.
- Subring
- 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.
- Rig
- 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".
- Rng
- 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".