Glossary of ring theory
From Wikipedia, the free encyclopedia
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.
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[edit] Definition of a ring
- Ring
- A ring is a set R with two binary operations, usually called addition (+) and multiplication (*), such that R is an abelian group under addition, a monoid under multiplication, and such that multiplication is both left and right distributive over addition. Note that rings are assumed to have multiplicative identities unless otherwise noted. The additive identity is denoted by 0 and the multiplicative identity by 1.
- 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.
[edit] 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 x of a ring is irreducible if it is not a unit and 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 it is not a unit and for any elements a, b ≠ 1 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.
[edit] 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 of (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
- The kernel is the preimage of the element 0 of the codomain of the ring homomorphism. Every ideal is the kernel of a ring homomorphism and vice versa.
- Maximal ideal
- A left ideal M of the ring R is a maximal left ideal if M ≠ R and the only left ideals containing M are R and M itself. Maximal right ideals are defined similarly. In commutative rings, there is no difference, and one speaks simply of maximal ideals.
- Nil ideal
- An ideal is nil if it consists only of nilpotent elements. A nil ring would be a ring consisting only of nilpotent elements; under the convention that rings are unital the only example would be the zero ring, but any rng (q.v.) that would be nil in that sense becomes a nil ideal inside the ring formed by adding 1. Nil algebras similarly.
- Nilpotent ideal
- An ideal I is nilpotent if its powers Ik are {0} for k large enough. Nilpotent implies nil, but not conversely. See the remarks at nil ideal for nilpotent rings or algebras. Locally nilpotent for a ring means finitely-generated subrings (subrings, really, since 1 is not required) are nilpotent.
- 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. There is also a definition of prime ideal for noncommutative rings.
- 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 prime 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
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- 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 monomorphism
- A ring homomorphism that is injective is a ring monomorphism.
- Ring epimorphism
- A ring homomorphism that is surjective is a ring epimorphism.
- Ring isomorphism
- 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.
[edit] 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. 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.
- Dedekind domain
- An integral domain in which every ideal has a unique factorization into prime ideals.
- Division ring or skew field
- A ring in which every nonzero element is a unit and 1≠0 is a division ring.
- Domain (ring theory)
- A ring without zero divisors and in which 1≠0. The noncommutative generalization of integral domain.
- 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 in fact an older branch of mathematics than ring theory.
- Integral domain or entire ring
- A commutative ring without zero divisors and in which 1≠0 is an integral domain.
- Invariant basis number
- A ring R has invariant basis number if Rm isomorphic to Rn as R-modules implies m=n.
- 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.
- Prime ring
- A common generalization of both simple rings and domains.
- Primitive ring
- A generalization of simple rings. Primitive rings are prime.
- Semisimple ring
- A ring that has a "nice" decomposition. A semisimple ring is also Noetherian, and has no nilpotent ideals. A ring can be made semi-simple if it is divided by its Jacobson radical.
- Simple ring
- A ring with no two-sided ideals.
- Unique factorization domain or factorial ring
- An integral domain R in which every non-zero non-unit element can be written as a product of prime elements of R.
- 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.
[edit] Miscellaneous
- Characteristic
- The characteristic of a ring is the smallest positive integer n satisfying n1 = 0 if it exists and 0 otherwise. As a consequence, nx=0 for all elements x of the ring.
- Direct product of a family of rings
- This is a way to construct a new ring from given rings by taking the cartesian product of the given rings and defining the algebraic operation component-wise.
- 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 divisors, and formally define their multiplicative inverses, which shall be added into R.
- 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".
